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III 

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AIR-SCREWS 

An   Introduction   to    the  Aerofoil   Theory  of 
Screw  Propulsion 


BY 

M.  A.  S.   RIACH 

)  i 

ASSOCIATE    FELLOW   OF   THE   AERONAUTICAL   SOCIETY 


**  In  mediis  qure  rigore  omni  vacant  resistantiiS  corporum 
stint  in  duplicata  ratione  velocitatum." — Newton. 


NEW    YORK 

D.   APPLETON    AND    COMPANY 
MCMXVI 


PRINTED    IN    CHEAT    liRITAIN 


PREFACE 

WITH  the  coming  of  the  Aeroplane  the  quantitative  study  of 
screws  working  in  air  has  assumed  a  great  importance. 

Formerly  in  the  design  of  screw  propellers  for  marine  work 
experiments  with  models  were  carried  out  and  the  performance 
of  the  full  size  screw  calculated  from  them,  and  it  was  not 
until  1882  that  Drzewiecki  first  drew  attention  to  a  possible 
more  powerful  method  of  design  obtained  by  considering  each 
element  along  the  blade  as  independent  and  behaving  in  the 
same  manner  as  if  moving  through  the  fluid  in  a  straight  line. 

This  method  has  since  assumed  great  importance  in  the 
practical  design  of  air-screw  blades,  and  the  results  obtained 
seemed  to  justify  the  utilization  of  this  theory  as  at  least 
approximately  correct  provided  certain  limits  are  not  exceeded. 

In  the  present  work  the  theory  has  been  assumed  to  be 
absolutely  correct,  and  the  results  obtained  have  been  carried 
to  their  logical  conclusions.  This  has  been  done  for  various 
reasons. 

It  does  not  make  for  completeness  in  any  argument  if  the 
results  of  the  initial  hypothesis  are  not  carried  to  their  ultimate 
logical  conclusions,  and  although  in  the  present  instance  the 
results  so  obtained  may  not  be  completely  borne  out  in  practice, 
yet,  in  giving  an  insight  into  the  applications  of  the  theory, 
and  in  establishing  at  any  rate  an  approximate  method  for 
dealing  with  the  many  cases  arising  out  of  the  performances 
of  aircraft,  the  conclusions  arrived  at  will,  it  is  hoped,  not  be 
without  interest. 

In  any  case  the  practical  application  of  some  of  the  more 
extreme  results  should  not  be  made  without  due  caution,  and 

358026 


iv  PREFACE 

in  fact  they  are  to  be  regarded  as  of  an  extremely  tentative 
nature.  This  caution  is  necessary,  for  "  There  is  no  more 
common  error  than  to  assume  that,  because  prolonged  and 
accurate  mathematical  calculations  have  been  made,  the  appli- 
cation of. the  result  to  some  fact  of  nature  is  absolutely  certain. 
The  conclusion  of  no  argument  can  be  more  certain  than  the 
assumption  from  which  it  starts  "  (Whitehead,  "  Introduction 
to  Mathematics ").  Mathematics  are  too  often  apt  to  be 
regarded  as  capable  of  "  creating  "  results,  quite  independently 
of  any  initial  hypotheses,  when  they  are  nothing  more  than  a 
very  useful  tool. 

I  have  endeavoured  to  present  the  subject  of  air-screw 
design  in  as  simple  a  manner  as  possible,  so  that  the  ordinary 
non-mathematical  reader  may  be  able  to  follow  the  train  of 
reasoning,  at  any  rate  as  far  as  its  qualitative  nature  is 
concerned. 

It  may  be  that  the  first  chapter  is  unnecessarily  drawn  out, 
but  it  appears  to  me  that  in  any  investigation  of  this  kind 
the  first  essential  is  to  be  able  to  clearly  ''visualise"  what  is 
being  done,  the  mere  application  of  analytical  processes  being 
but  a  secondary  matter. 

I  have  introduced  graphical  methods  wherever  it  seemed  to 
be  necessary,  or  where  it  was  impossible  to  obtain  solutions 
without  them.  At  the  same  time  the  results  for  the  design  of 
an  air-screw  to  fulfil  any  specified  outside  conditions  have 
been  put  into  such  a  form  that  it  is  hoped  that  the  design 
will  be  able  to  be  correctly  carried  out  by  the  rules  given, 
even  if  the  analytical  processes  have  not  been  able  to  be 
followed  by  the  reader. 

With  regard  to  the  possible  errors  involved  in  the  applica- 
tion of  the  results  given  in  the  text,  these  should  not  be  found 
to  materially  affect  any  but  the  last  chapters  of  the  book. 
The  chapters  on  "  static  "  thrust,  efficiency  of  air-screws  from 
(V)  equal  to  zero  up  to  the  velocity  of  flight,  and  on  direct 
lifting  systems,  are  admittedly  of  a  speculative  chara'cter. 


PREFACE  v 

It  did  not  seem  that  a  work  of  this  kind  could  be  regarded 
as  complete  without  some  reference  to  the  stresses  occurring  in 
an  air-screw  blade,  and  accordingly  a  chapter  on  centrifugal 
and  bending  stresses  has  been  included. 

It  is  hoped  that  the  book  will  be  found  to  be  not  without 
interest  to  engineers  desiring  an  introduction  to  the  theory  of 
air-screws,  while  at  the  same  time  it  may  perhaps  conduce  to 
a  more  scientific  study  of  the  subject,  in  place  of  what  has 
been  aptly  described  as  the  "  make  it  4  by  2 "  methods  so 
dear  to  the  heart  of  the  "  practical  "  man. 

I  take  this  opportunity  of  thanking  Mr.  H.  Bolas,  of  the 
Air  Department,  Admiralty,  for  Ids  valuable  criticisms  upon 
the  proofs.  My  thanks  are  also  due  to  Mr.  T.  E.  Ritchie  for 
his  help  in  reading  through  the  proofs,  and  to  Mr.  A.  King  for 
his  assistance  in  the  compiling  of  the  various  diagrams. 

The  Controller  of  His  Majesty's  Stationery  Office  has 
permitted  me  to  quote  from  certain  of  the  Technical  Reports  of 
the  Advisory  Committee  for  Aeronautics. 

M.  A.  S.  R. 
HENDON. 


CONTENTS 


INTRODUCTION. 

PAGE 

PRESSURE  ON  AEROFOILS  .  ...       1 


CHAPTER   I. 
THE  PITCH  OF  AN  AIR- SCREW        .......       6 

CHAPTER  II. 
THE  FORCES  ACTING  ON  AN  AIR- SCREW  BLADE       .          .         .         .14 

CHAPTER   III. 
BLADE  SHAPE  AND  EFFICIENCY       .......     25 

CHAPTER  IV. 
BLADE  SECTIONS,  AND  WORKING  FORMULAE     .         .         .         .         .47 

CHAPTER   V. 
"  LAYING  OUT  "  THE  AIR-SCREW    .         .         .         .         .         .         .72 

CHAPTER  VI. 
STRESSES  IN  AIR- SCREW  BLADES   ...  .77 

CHAPTER  VII. 
STATIC  THRUST     ....•••• 


viii  CONTENTS 

CHAPTER  VIII. 

PAGE 

EFFICIENCY  OF  AN  AIR- SCREW  AT  DIFFERENT  SPEEDS  OF  TRANSLATION     91 

CHAPTER  IX. 
DIRECT  LIFTING  SYSTEMS      ........   109 

APPENDIX   I. 
NOTE  ON  THE  INFLUENCE  OF  "ASPECT  RATIO"       ....   1'zi 

APPENDIX   II. 
NOTE    ON    THE    EFFECT    OF    THE    INDRAUGHT    IN     FRONT     OF    AN 

AlR-SCREW      .  .     123 


INDEX  .  .  .125 


AIE-SGEEWS 

INTRODUCTION. 

PKESSURE   ON   AEROFOILS. 

THE  problem  of  determining  the  form  of  air  flow  generated  by 
an  aerofoil,  moving  through  still  air,  is  one  that  has  so  far 
defied  mathematical  analysis.  This  problem,  which  applies 
equally  to  water  and  other  fluids,  is  one  for  which  a  solution 
has  only  been  obtained  when  the  aerofoil  is  a  flat  plate  of 
infinite  span.*  The  analytical  results  however  do  not  conform 
with  those  obtained  by  experiment,  and  it  would  seem  that  the 
still  more  complex  problem  of  the  curved  surface  is  beyond 
the  reach  of  present-day  analysis. 

The  mathematical  theory  makes  the  pressure  on  a  flat 
surface  vary  as  the  square  of  the  general  velocity  of  the  stream, 
and  experiment  has  shown  that  between  fairly  large  limits  in 
velocity  this  holds  good. 

Briefly,  if  an  aerofoil  be  exposed  to  a  moving  current  of  air 
or,  what  is  the  same  thing,  if  an  aeroplane  wing  be  moving 
through  still  air,  the  resultant  air  pressure  on  the  aerofoil  may 
be  expressed  by  the  formula  (R  =  &.S.V2),  where  (S)  denotes 
the  area  of  the  aerofoil  surface,  and  (V)  the  velocity  of  flight 
of  the  aerofoil  relative  to  the  air.  (k)  is  a  constant  the 
numerical  value  of  which  depends  upon  the  units  employed  in 
the  measurements  of  the  quantities  (R),  (S),  and  (V). 

*  A  further  development  has  recently  been  obtained  by  Professor 
G.  H.  Bryan  and  Mr.  E.  Jones.  "Discontinuous  Fluid  Motion  past  a 
Bent  Plane,  with  Special  Eeference  to  Aeroplane  Problems."  By  G.  H. 
Bryan,  Sc.D.,  F.E.S.,  and  Eobert  Jones,  M.A.  (Proceedings  of  the  Eoyal 
Society,  Vol.  91,  No.  A  630). 

B 


•2, :  :  . :    AIE-SCEEWS 

It  can  easily  be  seen  that  (k)  has  thus  the  dimensions  of 
density.  If  we  introduce  the  density  of  the  air  into  the 
equation,  we  can  write  (E  =  c./o.S.V2),  where  (p)  denotes 
atmospheric  density,  and  the  constant  (c)  is  then  non- 
dimensional. 

The  resultant  pressure  (E)  is  composed  of  two  quantities, 
the  pressure  on  the  under  surface  of  the  aerofoil  and  the 
negative  pressure  on  the  top  surface  of  the  aerofoil.  This 
latter  as  a  rule  forms  the  principal  portion  of  (E)  and  in  many 
cases  is  from  three  to  four  times  as  large  as  the  pressure  on  the 
under  surface. 

(E)  can  be  split  up  into  two  components,  measured  normal 
to,  and  tangential  with,  the  line  of  flight  of  the  aerofoil. 
These  two  components  are  termed  the  lift  and  drag  components 
respectively,  and  the  value  of  their  ratio  is  termed  the  lift- 
drag  ratio  of  the  aerofoil. 

Since  (c)  has  been  shown  to  be  non-dimensional,  the  two 
constants  in  the  expressions  for  the  lift  and  drag  will  also  be 
non-dimensional,  and  accordingly  we  may  write 

L  =  c^.p.S.V2, 
D  =  cx.p.S.V2, 

where  the  suffixes  (y)  and  (x)  affixed  to  (c)  denote  vertical  and 
horizontal  components  respectively. 

(cy)  and  (cx)  are  known  as  the  absolute  lift  coefficient  and 
absolute  drag  coefficient  respectively. 

From  the  above  it  can  be  seen  that  the  ratio  ^-  is  equal 

to  the  ratio  — ,  and  so,  when  estimating  the  value  of  the  lift- 
cx 

drag  ratio  of  an  aerofoil,  it  is  unnecessary  to  know  the  value  of 
the  actual  pressures  concerned;  it  is  sufficient  to  know  the  value 
of  each  of  the  coefficients. 

f* 

The  value  of  this  ratio  -  is  of  great  importance  in  aeroplane 

cx 

design,  and  the  suitability  of  an  aerofoil  shape  as  a  wing  section 
depends  largely  upon  this  value. 

It  is  always  desirable  to  make  this  ratio  as  large  as  possible. 


INTRODUCTION  3 

This  also  applies  to  the  theory  of  air-screw  design  based  on  the 
aerofoil  analogy. 

In  records  of  experimental  results,  the  reciprocal  of  this 
ratio  is  sometimes  used. 

n 

The  ratio  -^  can  also  be  expressed  as  cot  7,  where  7  denotes 

cx 

the  gliding  angle  of  the  aerofoil. 

The  numerical  values  of  (cy)  and  (cx)  are  found  to  vary  with 
variations  in  the  angle  of  the  chord  incidence  of  the  aerofoil 
with  the  direction  of  motion,  and  their  values  are  usually  given 
over  a  large  range  of  angles. 

For  many  forms  of  aerofoil,  the  angle  corresponding  with 

the  highest  value  of  —  is  found  to  be  in  the  neighbourhood  of  4°. 

cx 

The  value  of  (c)  for  normal  incidence,  e.g.  "  broad  side 
on,"  of  a  flat  surface  of  infinite  span  is  (*44)  on  the  modern 
mathematical  theory,  the  actual  value  found  from  experiment 
being  ( •  64). 

There  is  another  quantity  upon  which  the  values  of  (cy) 
and  (cx)  depend  to  a  certain  extent.  This  quantity  is  the  ratio 
of  the  lengths  of  the  span  to  the  chord  of  an  aeroplane  wing. 
It  is  commonly  known  as  the  "  Aspect  Katio." 

Model  aerofoils  used  for  purposes  of  testing  in  a  wind 
tunnel  usually  have  a  value  of  (6)  for  this  ratio.  As  a  rule 
the  higher  the  value  of  the  aspect  ratio,  the  higher  the 

value  of  —  . 

C* 

It  can  hardly  be  said  that  there  is  anything  essentially  new 
in  the  development  of  a  theory  of  air-screws  from  the  analogy 
presented  with  the  motion  of  an  aerofoil  in  a  straight  line. 

This  subject  was  first  investigated  by  Drzewiecki  in  1882 
and  has  since  been  developed  by  Lanchester.* 

The  fundamental  hypothesis  underlying  the  whole  theory 
here  set  forth  is  that : 

(1)  Each  infinitesimal  element  along  the  blade  of  an  air- 
screw may  be  treated  as  a  separate  aerofoil  possessing  the  same 
characteristics  as  those  which  an  aerofoil,  having  the  same  shape 

*  "Aerial  Flight,"  by  F.  W.  Lanchester. 

B   2 


4  AIR-SCREWS 

and  haviDg  an  aspect  ratio  equal  to  the  aspect  ratio  of  the 
whole  air-screw,  would  possess  ; 

(2)  The  velocity  of  each  blade  element,  compounded  of  the 
translational  velocity  and  the  circumferential  velocity,  at  the 
point  in  question,  may  be  treated  as  causing  no  appreciable 
variation  from  the  V2  law,  so  that  the  infinitesimal  pressures 
on  every  element  of  the  blade  may  be  considered  as  satisfying 
the  relation  K  =  c.p.S.V2. 

Messrs.  F.  H.  Bramwell  and  A.  Fage,  of  the  National 
Physical  Laboratory,  say  with  reference  to  the  application  of 
these  assumptions  : — * 

"  There  are  at  present  two  systems  mainly  employed  in  the 
design  of  propellers.  The  first,  which  is  generally  used  in 
the  design  of  marine  propellers,  consists  in  making  small 
variations  from  existing  successful  designs :  it  is  necessary 
that  no  very  great  variation  should  be  made  at  any  one  time. 

"  The  second,  which  has  so  far  been  used  almost  exclusively 
for  the  design  of  aerial  propellers,  attempts  to  predict  the 
performance  of  a  propeller  from  a  consideration  of  the  forces 
on  elementary  strips  of  the  blade.  This  method,  if  sensibly 
correct,  is  far  more  powerful  than  the  older  one,  as  it  affords 
a  means  of  introducing  new  features  irrespective  of  whether 
the  variation  from  existing  types  is  small  or  large. 

"  The  initial  assumptions  underlying  this  method,  which  has 
been  developed  by  Lanchester  and  Drzewiecki,  are  that  the 
forces  on  the  blades  are  due  directly  to  the  velocities  of  the 
various  sections  relative  to  still  air,  these  velocities  being  com- 
pounded of  the  translational  velocity  and  the  circumferential 
velocity  at  the  point  in  question,  and  also  that  the  sections 
may  be  treated  independently.  .  .  . 

"  The  final  conclusion  arrived  at  is  that  although  the  method 
is  not  strictly  correct,  yet  in  the  hands  of  a  careful  designer  it 
is  probably  by  far  the  best  method  that  can  be  used  for  the 
design  of  propellers  in  the  present  state  of  knowledge  on  the 
subject.  .  .  . 

"  The  question  of  practical  importance,  however,  is  whether 

*  "Technical  Eeport  of  the  Advisory  Committee  for  Aeronautics, 
1912-13." 


INTRODUCTION  5 

the  theory  affords  a  sufficient  basis  for  purposes  of  design. 
When  examined  from  this  point  of  view  it  is  founcf  that,  if 
the  range  of  comparison  be  limited  to  that  usual  in  flying 
machines,  the  experimental  and  calculated  results  are  in 
sufficiently  good  agreement,  and  that  so  long  as  tho  conditions 
under  which  the  propeller  is  working  are  not  varied  too  widely, 
the  theory  may  be  satisfactorily  applied.  The  occasional  failure 
of  propellers  to  satisfy  the  conditions  of  design  may  be  due  to 
an  overstepping  of  these  limits.  In  most  cases  the  differences 
between  the  calculated  and  experimental  results  are  not 
sufficiently  large  for  their  effects  to  be  observed  in  the  flying 
of  the  completed  aeroplane.  .  .  . 

"  On  the  other  hand,  it  is  more  probable  that  most  of  the 
discrepancies  are  due  to  the  centrifugal  forces  on  the  air  in 
contact  with  the  blade ;  this  must  alter  the  character  of  the 
flow  round  the  blade  very  considerably,  and  it  is  perhaps  a 
matter  for  surprise  that  the  agreement  between  the  calculated 
and  experimental  values  is  as  close  as  it  is,  and  not  that  they 
do  not  agree  exactly." 


AIE-SCEEWS 


CHAPTER  I. 

THE  PITCH   OF  AN   AIR-SCREW. 

IT  is  fairly  obvious  that  when  an  air-screw  is  moving  through 
the  aii1  with  some  definite  translational  velocity  the  distance  it 
traverses  in  each  revolution  will  be  constant,  providing  the 
line  of  flight  be  horizontal. 


FIG.  l. 

Now  if  we  consider  any  portion  of  the  blade  at  a  distance 
of  (x)  feet  say  from  the  centre  of  the  boss  of  the  air- screw,  we 
shall  find  that  as  the  air-screw  as  a  whole  moves  forward,  the 
portion  of  the  blade  under  consideration  moves  up  some 
particular  helicoidal  path  due  to  the  air-screw  rotating  about 
its  axis  at  the  boss  centre. 


THE  PITCH  OF  AN  AIR-SCREW 


The  steepness  of  the  helix  traversed  by  the  portion  of  the 
blade  at  radius  (x)  will  depend  upon  the  value  of  the  distance 
traversed  translationally  by  the  air-screw  in  each  revolution. 

It  will  also  depend  upon  the  value  of  (#),  that  is  upon  the 
distance  of  the  portion  of  the  blade  considered  from  the  centre 
of  the  boss  of  the  air-screw. 

This  may  perhaps  be  more  clearly  visualized  if  we  imagine 


FIG.  2. 

a  cylinder  having  a   radius  of  (x)  and  a  depth  of  (P),  as  in 

Kg.  (1). 

Or  it  may  be  demonstrated  by  taking  a  rectangular  piece  of 
paper  and  drawing  a  diagonal  line  as  in  Fig.  (2). 

Then  if  the  paper  be  rolled  so  as  to  form  a  cylinder  having 
both  ends  open,  the  diagonal  line  will  represent  the  path  or 
helix  traversed  by  the  point  under  consideration ;  the  diameter 
of  the  cylinder  so  formed  will  represent  twice  the  distance  of 
the  point  considered  from  the  centre  of  the  cylinder  (i.e.  the 
centre  of  the  boss  of  the  air-screw). 

The  circumference  of  the  base  of  the  cylinder  will  represent 


8 


AIE-SCEEWS 


the  path  that  would  be  traced  out,  by  the  point  considered,  in 
one  revolution,  if  the  air-screw  was  not  moving  forward  at  all, 
but  merely  revolving  on  its  axis  ;  and  the  length  of  this  circum- 
ference can  be  seen  to  be  equal  to 

(Tr).(the  diameter  of  the  cylinder), 

that  is  (2.7T.X.),  where  (x)  is  the  distance  of  the  walls  of  the 
cylinder  from  the  centre  of  the  cylinder,  corresponding  to  the 


K*) 


2TTX 


FIG.  3. 

distance  of  the  portion  of  the  air-screw  blade  considered  from 
its  boss  centre. 

And  the  depth  of  the  cylinder  will  then  represent  the 
distance  advanced  through  translationally  by  the  point,  and 
therefore  by  the  whole  air-screw,  at  each  revolution. 

If  the  paper  cylinder  be  now  flattened  out  as  in  (Fig.  3),  it 
at  once  becomes  apparent  that  the  helix  traversed  by  the  point 
in  each  revolution  of  the  air-screw  is  the  hypotenuse  of  a  right- 
angled  triangle,  and  therefore  that  its  length  is  equal  to 
2"^  b  Euclid  L  4Hr 


THE  PITCH  OF  AN  AIR-SCREW  9 

Let  us  denote  the  angle  which  the  helix  line  makes  with 
the  base  by  (A)°.  Then  if  (V)  ft./sec.  be  the  translational 
velocity  and  (n)  the  number  of  revs.  /sec.  of  the  air-screw,  we 
see  that  the  distance  advanced  in  each  revolution  translationally 


y 

is  —  feet. 


This  distance  we  shall  here  denote  by  (P),  and  (P) 


is  then  defined  as  being  the  effective  pitch  of  the  air-screw. 

We  have  so  far  only  considered  the  path  traced  out  by  one 
portion  of  the  blade  at  a  radius  of  (x).  Suppose  that  we  have 
a  large  number  of  paper  cylinders,  each  having  the  same  depth 
but  of  varying  diameters.  Then  their  respective  diagonals 


will  represent  the  respective  helicoidal  paths  traced  out  by  the 
various  portions  of  the  blade.  It  is  at  once  obvious  that  to 
completely  portray  the  paths  traced  out  by  every  portion  of 
the  blade  we  should  require  an  infinite  number  of  such 
cylinders. 

We  can,  however,  do  this  very  easily  for  a  limited  number 
of  different  parts  of  the  blade  by  drawing  a  right-angled 
triangle  having  a  base  equal  to 

(2.7r).(half  the  diameter  of  the  air-screw), 

and  having  straight  lines  drawn  from  the  vertex  of  the  triangle 
to  various  points  along  the  blade,  as  in  Fig.  (4). 

The  height  of  the  triangle  is  then,  as  before,  equal  to  (P), 


10  AIR-SCEEWS 

the  effective  pitch,  and  the  angles  formed  by  the  various  lines 
at  the  base  of  the  triangle  represent  the  helix  angles  of  the 
respective  portions  of  the  blade  considered. 

We  may  denote  these  angles  by  (A1}  A2,  A3, )  corre- 
sponding to  radii  from  the  boss  centre  of  (xlt  x2,  x3 ). 

The  helix  angle  of  the  blade  tip,  that  is  the  angle  at  a 
radius  equal  to  half  the  diameter  of  the  air-screw,  may  be 
denoted  by  (0). 

It  can  be  seen  from  the  figure  that  since  all  the  helicoidal 
paths  of  the  various  portions  of  the  blade  meet  in  a  point,  they 
must  all  satisfy  the  relation 


tan  A  =  -., 


p  /    p    \ 

— ,  so  that  A  =  tan-1  (  ^ ), 

2.7T.X  \2.7T.xJ 


giving  the  value  of  the  helix  angle  (A)  in  degrees  for  any 
radius  (x)  from  the  boss  centre  of  the  air-screw,  providing  the 
value  of  (P)  be  known. 

Now  suppose  that  we  have  an  air-screw,  and  that  we 
measure  the  actual  chord  angles  of  the  blade  to  the  disc  of 
revolution  at  various  distances  from  the  boss  centre. 

Let  us  denote  these  various  chord  angles  by  (fa,  fa,  fa ), 

corresponding  to  radii  of  (xl}  x2,  x3 )  from  the  centre  of 

the  boss. 

In  some  types  of  air-screws,  these  angles  are  designed  so 
that  they  all  satisfy  the  relation 


(<£)  being  any  chord  angle  measured  along  the  blade.* 

Air-screws  of  such  a  type  are  sometimes  said  to  be  of 
"  constant  pitch,"  although  this  term  is  somewhat  misleading. 
It  must  not  be  confused  with  the  "effective  pitch"  already 
defined. 

The  above  condition  of  an   air-screw  having   a    "constant 
pitch  "  is  however  in  the  nature  of  a  restriction,  and  we  shall 

y 

*  (P)  does  not  necessarily  here  denote  the  value  of  — ,  but  usually  has 

a  larger  value  than  the  effective  pitch  of  the  air-screw. 


THE  PITCH  OF  AN  AIE-SCEEW  11 

therefore   start  with   the   assumption   that   the   chord   angles 

($i»  4>z,  <f>3 )  have  no  specified  connection,  but  that  they 

may  be  anything  whatever. 

Now    suppose    our    air-screw,    having    the    chord    angles 

(<£i>  $2>  $3 )  as  defined,  to  have  a  translational  velocity 

of  (V)  feet/sec,  and  a  speed  of  revolution  of  (ri)  revs. /sec.,  then 
the  distance  advanced  through  in  the  direction  of  translation 

y 
per  revolution  is  -  -  feet,  and  has  already  been  defined  as  being 

the  effective  pitch  of  the  air-screw. 

Suppose  that  we  keep  (n)  constant,  and  give  to  (V)  the 
successive  values  of  (V1}  V2,  V3 ). 

Then   the  distances  advanced  through  at  each  revolution 

.n  ,     VT  V2  V3 
will  be—  ,  — ,  — 

n      n     n 

Thus  it  is  possible  to  have  an  infinite  number  of  values 
for  the  effective  pitch  of  any  given  air-screw,  correspond- 
ing to  an  infinite  number  of  values  of  the  translational 
velocity  (V). 

It  is  obvious  therefore  that  the  effective  pitch  of  any  air- 
screw is  not  necessarily  a  fixed  quantity,  but  depends  upon  the 
values  of  (V)  and  (n). 

It  is  usual  however  to  associate  the  pitch  of  a  screw  with 
the  screw  itself,  and  thus  to  imagine  it  to  be  a  fixed  quantity 
for  any  given  screw. 

In  order  to  determine  some  analogous  expression,  in  the 
case  of  an  air-screw,  which  is  a  constant  quantity  for  any  given 
type  of  air-screw,  we  may  define  a  particular  value  of  the  ratio 
y 

-  at  which  there  is 
n 

(1)  no  resultant  thrust  on  the  blades  in  the  direction  of 

translation ; 

(2)  no  resultant  torque  on   the   blades  in  a  direction 

normal  to  the  direction  of  translation,  and  there- 
fore tangential  to  the  disc  of  revolution  of  the 
blades ; 

•  (3)  no  " average"  reaction  on  the  blades. 


12 


AIE-SCKEWS 


The  three  values  of  — 'corresponding  to  these  three  cases 

will  be  constants  for  any  given  type  of  air-screw. 

The  quantitative  determination  of  these  values  of  the 
effective  pitch  may  be  found  from  the  results  of  the  analysis 
to  be  proved  later. 

We  may  denote  these  values  of  (P)  by  (Pj),  (P2),  and  (P3), 

or  by  P),P),  and  (I). 
\n/i  W/2          \n/3 

The  "experimental  mean  pitch"  of  an  air-screw  has  been 


FIG.  5. 


defined  by  Mr.  F.  H.  Bramwell  as  being  the  value  of  the  ratio  - 

TI 

for  which  there  is  no  thrust  on  the  blades  in  the  direction  of 
translation. 

This  definition  corresponds  to  (1)  already  given. 

Suppose  then  that  our  air-screw  receives  a  velocity  of  trans- 
lation of  (V)  feet/sec.,  and  therefore  has  some  definite  value  of 
y 

— ,  the  effective  pitch. 
n 


THE  PITCH  OF  AN  AIK-SCREW  13 

,  Denoting  the  helix  angles  along  the  blade  at  radii  of 

(xlf  a?a,  a?3,  )  by  (Alf  A2,  A3,  ......)  and  the  chord 

angles  by  (<j>lt  $2,  </>3, )  we  may  represent  the  paths  of 

the  various  portions  of  the  blade  considered  by  Fig.  (5). 

We  have  made  the  chord  angles  in  each  case  greater  than 
the  corresponding  helix  angles  for  the  sake  of  clearness. 

Then  it  is  obvious  that,  since  each  of  the  blade  elements 
considered  is  moving  up  the  hypotenuse  of  one  of  the  corre- 
sponding right-angled  triangles,  the  actual  angles  at  which 
these  blade  elements  move  to  their  respective  helicoidal  paths 

are  (<£i  -  Aj),  (<£2  -  A2),  (<£3  -  A3), ,  and  these  angles 

are  analogous  to  the  chord  angles  of  incidence  of  an  aeroplane 

wing  in  flight.  They  are  here  denoted  by  (al}  a2,  a3, ) 

and  are  termed  the  "angles  of  attack"  of  the  various  blade 
elements  considered. 

Since  the  angles  (<£lf  <f>2,  <£3, )  are  perfectly  arbitrary, 

these  angles  of  attack  are  likewise  arbitrary  and  may  be  made 
anything  that  is  convenient. 


14  AIK-SCEEWS 


CHAPTEE  II. 

THE   FORCES  ACTING  ON  AN  AIR-SCREW   BLADE. 

Now  it  has  already  been  stated  that  if  an  aerofoil  be  moving 
in  a  straight  line  at  a  velocity  of  (V),  the  two  components, 
normal  to  and  tangential  with  its  line  of  flight,  of  the  resultant 
pressure  exerted  upon  it  by  the  air  can  be  written 

L  =  cy./>.S.V2, 
D  =  ^.p.S.V2, 

and  we  have  the  obvious  relation  already  mentioned, 
L       cy 

D  =  ;r  =  cot* 

(7)  being  the  gliding  angle  of  the  aerofoil  at  the  particular 
angle  of  incidence  to  its  line  of  flight  considered. 

(7)  is  the  angle  which  the  direction  of  the  resultant 
air-pressure  on  an  aerofoil  makes  with  the  direction  of  the 
vertical  component  of  (R),  that  is  the  lift. 

It  will  be  noticed  that  the  blade  sections  of  an  air-screw 
are  similar  to  those  used  on  aeroplane  wings,  and  hence  it  at 
once  raises  the  question:  Cannot  we  treat  the  sections  along 
an  air-screw  blade  as  if  they  were  aeroplane  wings  moving  at 
angles  of  incidence  of  (alf  a2>  «s> )  ? 

It  is  upon  this  very  assumption  that  the  whole  of  the 
theory  of  air-screw  design  is  based,  as  already  explained  in  the 
Introduction. 

It  is  also  obvious  that,  in  order  to  be  able  to  correctly 
follow  out  the  consequences  of  this  assumption,  we  must  treat 
each  of  the  portions  of  the  blade  as  being  of  infinitely  small 
oreadth  in  the  direction  of  the  radius  (x). 


FOBCES  ACTING  ON  AN  AIE-SOEEW  BLADE     15 

We  may  then  without  error  sum  up  the  forces  on  all  these 
infinitely  narrow  blade  strips,  and  hence  obtain  a  correct 
quantitative  determination  of  the  characteristics  of  the 
air-screw.* 

This  is  of  course  the  ordinary  mathematical  process  of 
integration. 

Let  us  then  consider  the  forces  acting  upon  a  strip  of  blade 


FIG.  6. 

at  a  radius  of  (x)  from  the  boss  centre  of  the  air-screw. 
Fig.  (6)  shows  the  two  views  of  the  blade,  in  side  elevation 
and  plan. 

Since  the  element  of  blade  is  moving  up  the  hypotenuse  of 
the  right-angled  triangle,  the  two  components  of  the  resultant 

*  This  is  of  course  an  assumption  of  the  theory. 


16  AIE-SCKEWS 

air-pressure  upon  it  will  be  normal  to  and  tangential  with  the 
hypotenuse  of  the  triangle  respectively. 

These  infinitesimal  pressures  may  be  denoted  by  (dL) 
and  (dD). 

Let  the  width  of  the  blade  at  radius  (x)  be  denoted  by  (&). 
Then,  applying  the  formulae  for  air-pressure,  we  have 

dL  =  Cy.p.b.dx.v2, 
and  dD  =  cx.p.b.dx.v2. 

And  the  value  of  (v)  is  obviously  equal  to 


for,  as  already  shown,  the  quantity  ^P24-4.7r.V  is  the  length 
of  the  helix  traversed  by  the  element  of  blade  at  each  revolution 
of  the  air-screw,  and  hence  the  distance  traversed  by  the  element 
per  second  is  (n)  times  this  amount,  where  (n)  is  equal  to  the 
number  of  revolutions  of  the  air-screw  per  second. 

Whence  the  velocity  of  the  blade  element  is  n.  x/PM-^Tr2^2. 
So  that  we  have 

dL  =  p.ri*.Cy.1>.  (P2  +  4.7r2.£2).  dx, 
dD  =  p.n'2.cx.b.  (P2  +  4.7r2.z2).  dx. 

Now  we  are  not  immediately  concerned  with  (dL),  but  with 
its  components  normal  to  and  tangential  with  the  disc  of 
revolution  of  the  blade. 

These  are 

(dL.  cos  A)  and  (dL.  sin  A)  respectively. 

Again,  consider  the  value  of  (dD),  the  drag  of  the  element. 
We  have 

dD  =  P.n2.cx.b.  (P2  +  4.7i2.#2).  dx, 

and  this  may  also  be  split  up  into  two  components  normal  to 
and  tangential  with  the  disc  of  revolution  of  the  blade. 
These  components  are 

(  —  dD.  sin  A)  and  (dD.  cos  A)  respectively. 

Now  the  thrust  on  the  element  is  measured  by  the  com- 
ponents of  all  the  forces  acting  on  the  element  in  a  direction 


FOBCES  ACTING  ON  AN  AIR-SCEEW  BLADE     17 

normal  to  the  disc  of  revolution  of  the  blade,  that  is,  parallel 
with  the  line  of  advance  of  the  air-screw. 
The  forces  so  acting  are  seen  to  be 

(dL.  cos  A)  and  (  —  dD.  sin  A), 

so  that  the  resultant  force  on  the  element  in  a  direction  normal 
to  the  disc  of  revolution  of  the  air-screw  is  (dL.  cos  A  —  dD.  sin  A), 
and  this  may  be  denoted  by  (dT),  the  elementary  thrust  on  the 
element. 

In  a  similar  manner  it  can  be  shown  that  the  remaining 
forces  make  up  a  total  force  in  a  direction  tangential  to  the 
disc  of  revolution  of  the  air-screw,  and  of  amount 

(dL.  sin  A  4-  dD.  cos  A). 

This  force  comprises  the  drag  or  resistance  exerted  by  the 
element  to  circular  motion,  and  is  analogous  to  a  friction 
brake  applied  to  the  rim  of  a  revolving  wheel.  It  tends  to 
retard  the  motion. 

It  can  be  seen  that  the  product  of  the  above  quantity  and 
the  distance  (x)  of  the  element  from  the  boss  centre  measures 
the  torque  of  the  element. 

Thus,  denoting  the  torque  by  (dM.),  we  have 

dM  =  x.  (dL.  sin  A  +  dD.  cos  A), 

and  we  have  already  shown  that  the  thrust  on  the  element  may 
be  expressed  by 

dT  =  (dL.  cos  A-dD.  sin  A). 

From  these  two  equations  and  from  the  equation  already 
obtained  for  the  lift  on  the  aerofoil,  we  can  solve  all  the 
problems  presented  in  the  design  of  air-screws. 

We  have 

dT  =  dL.  cos  A  —  dD.  sin  A, 


and  we  know  that 


cx       drag       dD 

tan  i  =     =:  "Eft-  =  dL' 


18  AIPt-SCREWS 

whence,  substituting,  we  obtain 

dTl  =  p.n2.Cy.b.  (2.7r.x  —  P.  tan  7).\/P2-f-4.7r2..?j2.  dx, 
and  therefore 


T  =  p.n2.       A  (2.7r.a;-P.  tan  7)VF+47T2^?.  dx, 
Jr, 

giving  the  total  thrust  on  each  blade  of  the  air-screw. 

(r)  denotes  the  length  of  each  blade  from  the  boss  centre. 

(r0)  denotes  the  length  from  the  boss  centre  to  the  extreme 
portion  of  the  blade  where  the  "  streamlining  "  of  the  sections 
ceases.  As  a  rule,  in  determining  approximate  values  for  the 
integrals,  we  may  take  (?10)  as  equal  to  zero  without  appreciable 
error. 

We  have  also 

dM  =  x.  (dL.  sin  A+d~D.  cos  A) 
=  p.tf.Cy.l.x.  (P  +  2.7r.x.  tan 

and  therefore 


M  =    .?i2.  \C.b.x.  P  +  2.7r.#.  tanVP2  +  4.7T2.£2.  dx. 


And  since  (M.2.7rji)  is  proportional  to  the  B.H.P.  required 
to  turn  each  blade  of  the  air-screw,  we  can  at  once  determine 
the  necessary  value  of  the  blade  width  at  each  radius  to  satisfy 
any  given  set  of  conditions. 

We  are  now  in  a  position  to  write  down  the  efficiency  of 
the  whole  blade,  that  is,  of  the  air-screw. 

The  work  done  per  revolution  by  the  air-screw  is  the 
product  of  the  total  thrust  exerted  and  the  distance  through 
which  it  is  exerted,  that  is,  the  quantity  having  the  value  of 

Y 

-  and  defined  as  the  effective  pitch  of  the  air-screw. 
n 

Let  (N)  denote  the  number  of  blades  of  the  air-screw. 
Then  we  have 

work  done^by  air-screw  per  revolution  =  (N.T.P.), 
and 

work  done  by  motor  per  revolution  in 

turning  the  air-screw  at  (n)  revs./sec.  =  (N.M.2.7T), 


FOKCES  ACTING  ON  AN  A  IE-SCREW  BLADE     19 

and  therefore  the  efficiency  of  the  whole  air-screw  is  given  by 
N.T.P. 


V  = 


and  /.  T  = 


N.M.2.7T. 
P.  I  l.Cy.  (2.7T.£-P.  tan  7)VFT4Pr.£2.  dx 

Jr0 


5.7T.      b.Cy.x. 

Jrn 


.7r2.a2.  dx 


This  gives  the  general  formula  for  the  efficiency  of  any  type 

V 
of  air-screw,  for  any  specified  value  of  —  . 

We  may  now  endeavour  to  determine  the  values  of  the 
pitch,  corresponding  to  no  thrust  on  the  blades,  to  no  torque 
on  the  blades,  and  to  no  resultant  "  average  "  reaction  on  the 
blades. 

The  first  of  these  quantities  is  called  the  "  Experimental 
Mean  Pitch,"  and  we  shall  now  endeavour  to  determine  by 
calculation  the  value  of  this  quantity  for  any  given  set  of 

conditions. 

y 
This  value  of  the  ratio        is  usually  determined  for  any 

given  air-screw  experimentally  in  a  wind-tunnel. 

The  method  to  determine  this  theoretically  is  as  follows. 

We  have  already  shown  that  the  expression  for  the 
thrust  on  each  blade  of  any  given  type  of  air-screw  may 
be  written  as 


T  =  p.n2.  \Cy.l.  (2.7T.&-P.  tan  7). 
Jr 


.  dx, 


Y 

where  P  =  —  . 
n 


Now  if  T  =  0,  then  since  (p.n*)  is  finite  we  have 

fr 

0  =      Cy.l.  (2.7r.a;-P.  tan  7).  \/P2+4.7ra.a;2.  dx, 

Jr0 


C   2 


20  AIR-SCREWS 

fr  _  _ 

i.e.  2.7T.  I  C.l).x.  \/P2-{-4.7r2.£2.  dx 


fr  _  _ 

2.7T.  I  Cy.l).x.  \/P2-{-4.7r2.£2. 
Jr0 

=  P.     cy.t>. 
Jr 


tan  7.  \P2  +  4.7r2.£2.  dor. 


And  this  may  be  solved  graphically  as  follows. 

Take  successive  values  of  (P)  from  some  value  greater  than 
what  would  correspond  to  the  effective  pitch  of  the  air-screw 
upwards,  and  plot  the  two  graphs  2.7r.%.6.#.\/P2-{-4.7r2.a;2  and 
P.Cy.6.  tan  7.  \/P2  +  4.7r2.a32  against  (x)  between  (?•„)  and  (r)  for 
each  of  the  values  of  (P)  taken.  Take  the  areas  of  the  two 
figures  thus  obtained  in  each  case.  When  the  two  areas  in  any 
case  are  numerically  equal,  then  the  value  of  (P)  taken  for  this 
case  is  the  value  of  the  "  experimental  mean  pitch  "  of  the  air- 
screw. 

It  is  of  course  apparent  that  the  values  of  (cy)  and  (tan  7) 
will  vary  with  each  value  of  (P)  taken. 

We  can,  however,  determine  their  respective  values  in  the 
following  manner. 

Let  us  denote  the  successive  arbitrary  values  of  (P)  taken 
by  (P',  P",  P'",  ......  ),  and  let  us  suppose  that  P'  <  P"  <  P'"  < 


Moreover,  let  us  denote  the  chord  angles  along  the  blade  of 

the  air-screw  by  ($!,   <£2,  $3 ).     Then  if  (A/,  A2',  A3' 

)    be    the    helix    angles    at    radii   (xit   x2,   x3 ) 

y 
respectively,  for  a  value  of  —  equal  to  the  first  value  of  (P) 

taken,  namely  (P'),  the  value  of  each  of  these  helix  angles  at 
any  radius  (x)  will  be  given  by  the  relation 

A'  =  tan- 


2.7T.O?./ 

And  therefore  the  values  of  the  respective  angles  of  attack 
of  the  sections  at  these  radii  will  be  given  by 

a  =  (<j>  —  A')  =  <£  —  tan"1 
for  the  particular  value  (Pf)  considered. 


FOKCES  ACTING  ON  AN  AIE-SCEEW  BLADE     21 

Similarly,  it  may  be  shown  that  the  angles  of  attack  of  the 
sections  for  the  other  values  of  (P)  taken,  namely  (P",  P"' 
),  will  be  given  by 


'" 
a'"  =     - tan-1      - 


Now  measure  up  the  forms  of  the  sections  of  the  blade  at, 
say,  (8)  different  radii.  Then,  if  the  sections  so  obtained  are 
ones  that  have  been  tested  as  aerofoils  in  a  wind-tunnel,  we 
can  at  once  write  down  their  characteristics  for  any  value  of 
the  angle  of  attack  (a). 

We  may  thus  plot  a  series  of  graphs,  for  each  of  the 
arbitrary  values  of  (P)  taken,  of  (cy)  and  (tan  7)  for  the 
different  radii  taken  along  the  blade. 

If  we  then  construct  a  series  of  such  graphs  for  (cy)  and 
(tan  7)  for  all  the  values  of  (P)  taken,  namely  (P,  P",  P'" 

),  we  shall  be  in  a  position  to  determine  the  value  of 

the  experimental  mean  pitch. 

And  we  have  to  plot  the  two  graphs  already  given  for  all 
the  values  of  (P)  taken. 

In  order  to  be  able  to  plot  these  two  curves  in  any  case, 
we  must  draw  out  a  curve  of  the  actual  blade  widths  against 
radii  (#). 

It  then  only  remains  to  obtain  the  two  areas  enclosed  by 
these  two  curves  and  the  (x)  axis.  The  value  of  (P)  taken, 
which  makes  these  two  areas  equal,  is  the  value  of  the  experi- 
mental mean  pitch,  corresponding  to  no  thrust  on  the  blade. 

y 

In  a  similar  manner  we  can  determine  the  value  of  —  for 

n 

which  there  is  no  torque  on  the  whole  blade. 

V 

To  determine  the  value  of  --  for  which  there   is   a   zero 

n 

value  of  the  "  average  "  reaction  over  the  whole  blade,  it  is  first 
necessary  to  investigate  the  value  of  this  resultant  pressure  at 
any  radius  (x),  and  so  compute  the  value  over  the  whole  blade 
of  the  air-screw. 


22 


AIR-SCREWS 


We   shall   then   be   in   a   position   to   estimate   the   value 
y 
of  -  -  for  which   this  "  average "  resultant   pressure   over  the 

whole  blade  is  zero. 

In  order  to  obtain  the  value  of  this  resultant  pressure  (R) 
we  proceed  as  follows. 

Consider  a  blade  section  at  radius  (#),  and  having  a  helix 
angle  of  (A),  Fig.  (7). 

The  resultant  air-pressure  (R)  upon  the  section  is  usually 


T 


FIG.  7. 


split  up  into  two  components  called  the  lift  and  drag  of  the 
section.     Let  (7)  denote  the  gliding  angle  of  the  section. 
Then 

Lift    =  R.  cos  7, 
and 

Drag  =  R.  sin  7, 
whence 

Lift 

=  cot  7. 


Now  we  have  already  shown  that  the  resultant  thrust  on 
the  section,  in  a  direction  parallel  to  the  line  of  advance  of  the 
whole  air-screw,  is  given  by 

T  =  L.  cos  A-D.  sin  A, 


FOECES  ACTING  ON  AN  AIE-SCKEW  BLADE     23 

where  (L)  and  (D)  denote  the  lift  and  drag  components  of  (R) 
respectively. 
But 

L  =  R.  cos  7, 
and 

D  =  R.  sin  7, 
whence 

L.  cos  A  —  D.  sin  A  =  R.  cos  7.  cos  A  —  R.  sin  7.  sin  A, 

and  therefore 

T  =  R.  cos  (A  +  7), 
whence 

R  =  T.  sec  (A +  7)     for  every  point  along  the  blade. 

It  is  also  obvious  that 

1 

sec  (A +7)  =  -  — T—  — : — 7—^ — 

cos  A.  cos  7  —  sin  A.  sm  7 


2.7T.OJ.  cos  7  —  P.  sin  7' 
whence 

T.   \/P2+4.7r2.a;2 

~  cos  7.  (2.7r.a-P.  tan  7)' 
and  this  in  strictness  should  be  written 


~  cos  7.  (2.ir".i-P.  tan  7)' 
since  the  pressures  considered  are  infinitesimals/ 


And  dT  =  p.n2J>.Cy.  (2.7r.aj-P.  tan  7). 
so  that 

^R  =  p.?ia.6.cy.  (P2+4.7ra.aj2).  sec  7.  ^, 
whence 

R  =  p.n\  Ib.Cy.  (P2  +  4.7r2.^2).  sec  7.  dx, 

Jro 

and  if  this  be  now  plotted  against  (x),  we  shall  obtain  the 
"  Load  Grading  Curve  "  for  the  whole  blade. 

Since  (tan  7)  is  usually  nearly  constant  between  (r)  and 


24  AIE-SCEEWS 

(r0),  and  also  of  small  amount,  we  may  write  (sec  7)  equal  to 
unity,  and  our  expression  for  the  total  resultant  force  over  the 
whole  blade  then  becomes 


fr 

R  =  p.w?.  I  b.Cy. 

Jr0 


dx, 


and  the  curve  obtained  when  integrating  this  expression 
graphically  will  not  appreciably  differ  from  the  one  obtained  by 
inserting  the  rather  troublesome  (sec  7)  under  the  integral  sign. 

Having  then  obtained  the  value  of  (R)  for  each  blade  of  the 
air-screw,  we  may  proceed  to  determine  the  value  of  (P)  for 
which  (B)  has  a  zero  value,  in  the  same  manner  as  for  the  case 
of  a  zero  value  of  the  thrust  on  each  blade. 

y 

In  thus  estimating  this  value  of  —   it  will  be  prudent  to 

insert  the  (sec  7)  in  our  expression  for  (R). 
We  have  then 

R  =  p.n2.  I  b.Cy.  (P2  +  4.7T2.#2).  sec  7.  dx, 

Jr0 

and  since  (R)  =  0,  and  (p.n2)  is  finite,  we  obtain 

(r 

0  =   I  hep  (P2  -f  4.7r2.£2).  sec  7.  dx. 

J^o 


/       c  2         c. 2 
Now  (sec  7)  =  \/  I  -f-  — 2,  and  -^  is  always  a  positive  quantity. 

Cy  Cy 

Hence  (sec  7)  is  always  a  positive  quantity,  and  therefore 
(cy)  must  be  negative,  when  (R)  is  equal  to  zero,  for  at  least 
some  portion  of  the  blade. 


25 


CHAPTER   III. 

BLADE   SHAPE   AND   EFFICIENCY. 

WE   return   now   to   a   consideration  of  the  formula   already 
deduced  for  the  efficiency  of  any  type  of  air-screw  blade. 

It  is  obvious  that,  if  the  sections  at  every  radius  from  the 
boss  centre  along  the  blade  were  of  such  a  form  (if  it  were 

s* 

possible)  as  to  possess  no  drag,  so  that  the  value  of  —  would  be 

cx 

infinite,  the  efficiency  of  the  whole  air-screw  would  be  unity. 
That  this  is  so  may  be  seen  by  making  (tan  7)  equal  to  zero 
in  the  expression  already  obtained  for  the  efficiency  of  any 
type  of  blade. 
We  have 

P.  I  Cy.b.  (2.7r.x-l\  tan  7).  \/W+±^r\x\  dx 

n  -  Jr« 

V-  —rjr 

2.7r.      cy.b.x.  (P  +  2.TT.X.  tail  7).  \/P24-  4.7^.0?.  dx 

Jr« 

and  now,  making  (tan  7)  =  0,  we  get 
P.  I  Cy.b.2.7r  x. 


.7T. 


.  dx 


Unfortunately,  however,  (tan  7)  never  has  the  value  of  zero, 
although  the  smaller  its  value  the  higher  the  efficiency  and 
vice  versa. 

It  is  obvious,  since  (&)  is  a  function  of  the  radius  (x),  that 
the  value  of  (77)  will  vary  for  different  forms  of  blade  outline, 


26  AIPv-SCREWS 

? 

and  it  therefore  becomes  necessary^to  find  some  form  which 
will  give  as  large  a  value  as  possible  to  (?;)  consistent  with 
structural  considerations. 

Let  us  consider  the  efficiency  of  any  blade  element  at 
radius  (x)  from  the  boss  centre. 

We  have  already  shown  that  the  nett  thrust  of  the  element, 
in  a  direction  normal  to  the  air-screw's  disc  of  revolution,  is 
given  by 

dT  =  dL.  cos  A  (1  -tan  A.  tan  7), 

and  that  the  nett  drag  of  the  element,  in  a  direction  tangential 
to  the  disc  of  revolution  of  the  air-screw,  may  be  written 

c?R*  =  dlj.  sin  A-f-^D-  cos  A 

=  dL.  sin  A  (1  +  cot  A.  tan  7), 

and  therefore  the  work  done  by  the  element  per  revolution  of 
the  air-screw  is 

dT.P  =  P.</L.  cos  A  (1  -tan  A.  tan  7), 

and  the  work  done  by  the  motor  in  turning  the  element  per 
revolution  of  the  air-screw  is 

dP\,.2.Tr.x  =  2.7r.x.d~L.  sin  A  (1-fcot  A.  tan  7), 
whence  the  efficiency  of  the  element  is  given  by 

_  P.  cot  A.  (1  -  tan  A.  tan  7)  _       tan  A 
^A  ~       2.7r.x.  (1+cot  A.  tan  7)      ~  tan  (A  +  7)' 

where  the  suffix  (A)  in  (?7A)  signifies  the  efficiency  of  an  element 
at  a  helix  angle  of  (A). 

We  might  also  write  this  as  (rjx)  denoting  the  efficiency  of 
an  element  at  a  radius  of  (x).  So  that  (T;A)  =  (rjx). 

If  we  require  the  efficiency  of  the  element  in  terms-of  the 
radius  (x),  we  have 

P.  (2.7T.0-P.  tan  7) 
Vx  ~  Z.TT.X.  (P  +  2.7T.  tan  7.  x)' 

We  have  thus  shown  that  the  efficiency  varies  at  different 
points  along  the  blade,  and  hence  we  can  plot  a  curve  of  efficien- 
cies against  values  of  (x).  Such  a  curve  is  shown  in  Fig.  (8). 

*  Not  to   be  confused  with  the  resultant  pressure  on  an  element, 
denoted  by  the  same  symbol. 


BLADE  SHAPE  AND  EFFICIENCY  27 

The  maximum  point  of  efficiency  along  the  blade  may  be 
found  by  putting 

The  point  of  maximum  efficiency  is  found  to  be  at  a  value 

of  (A)  =  45°-?  that  is,  in  the  neighbourhood  of  43°. 

^  -* 

Since  tan  A  =  [  —^—  ),  we  can  write 

j.TT.XJ 

,   giving    the    value    of   (x)   for    the 


2.7T.  tan 
maximum  point  of  efficiency  along  the  blade. 


i'  a'  3'  4' 

Radius. (x) 
FlG.   8. 

The  point  of  maximum  efficiency  so  obtained  is  only  strictly 
true  when  (7)  is  supposecf  to  remain  constant  over  the  whole 
blade,  that  is,  for  every  radius. 

ft 

But  in  practice  the  values  of  -x-  for  the  various  blade  sections 

Cy 

will  not  be  found  to  vary  very  greatly,  and  hence  the  point  of 
maximum  efficiency  will  not  fall  very  far  short  of  that  given 
by  the  formula. 

It  is  obvious  from  the  foregoing  that  the  most  efficient  blade 
would  be  one  in  which  the  whole  of  the  blade  surface  was 
concentrated  at  the  point  of  maximum  efficiency.  The  blade 
width  (&)  would  then  become  infinite  and  the  length  of  the 
blade  in  the  direction  of  the  radius  (x)  would  be  infinitesimal. 


'28  AIB-SCKEWS 

As,  however,  such  a  form  of  blade  is  impossible  to  construct 
we  are  forced  to  adopt  a  compromise  between  the  width  of  the 
blade  and  the  diameter  of  the  air-screw. 

Let  us  first  consider  the  point  along  the  blade  at  which 
the  efficiency  reaches  a  maximum  value.  We  may  regard  (7) 
as  a  variable,  and  we  will  denote  it  by  tan"1  ty(x),  so  that 
tan  7  =  -ty(x)  ;  then  we  have 


2.TT.X. 

7  7O 

and  for  a  maximum  value  of  (%),-^  =  0,  and    ,  ^  is  negative. 

ClX  CLX 

So  that 


and  this  expression  gives  the  value*  of  (x)  for  which  (rjx)  is  a 
maximum. 

In  order  to  evaluate  the  above  it  is  necessary  to  know  the 
form  of  the  function  ty(x),  i.e.  (tan  7). 

If  (tan  7)  is  constant  over  the  whole  blade,  then  ^r'(x) 
vanishes,  and  the  above  expression  reduces  to 

P 


!.7r.tan(45°-|) 

We  thus  obtain  the  original  relation 
A  =  45°  —  - 

for  the  angle  at  which  the  point  of  maximum  efficiency  occurs 
along  the  blade. 

Now  we  have  not  as  yet  prescribed  any  particular  value 
to  (&),  the  width  of  the  blade  at  radius  (x). 

In  the  majority  of  air-screw  blades  the  value  of  (Z>)  varies 
for  different  values  of  (x),  i.e.  for  different  radii  along  the 
blade. 

Now  it  is  obvious  that  the  blade  should,  from  considerations 
of  overall  efficiency,  be  wider  at  some  radii  than  at  others. 
Hence  in  designing  our  blade  we  may  plot  a  curve  of  propor- 


BLADE  SHAPE  AND  EFFICIENCY  29 

tional  blade  widths  against  radii  (x\  so  that  the  true  or  actual 
blade  width  for  each  value  of  (x)  will  be  equal  to  the  width 
shown  on  the  curve  multiplied  by  some  constant. 

We  may  thus  write  the  blade  width  (&)  as  equal  to  the 
product  of  a  constant  and  a  function  of  (x),  or 

b  =  C.f(X). 

Now  we  have  already  assumed  that  (7)  is  a  variable  and 
may  have  different  values  for  different  values  of  (x),  and  we 

/> 

have  denoted  the  ratio  of  —  by  ^(x)y  so  that 

Cy 

tan  7  =  ty(x\     and  therefore 

7  =  tan"1  Tfr(x),     as  already  given. 

/i 

But  if  (7)  and  therefore  —  is  a  variable  or  function  of  (x), 

Cy 

so  also  is  (cy\  the  absolute  lift  coefficient  of  the  section  at 
radius  (x).  Hence  we  may  also  write  this  in  the  form 


It  has  already  been  shown  that 

P.      2.7T 


which  when  plotted  against  radii  (x)  will  give  a  curve  of 
efficiency  for  each  value  of  (x). 

Now  suppose  we  regard  the  curve  so  obtained  as  our  pro- 
portional blade  width  curve,  so  that  at  each  point  along  the 
blade  the  actual  blade  width  is  proportional  to  the  efficiency 
at  that  point.  We  shall  then  ensure  at  any  rate  that  our  blade 
is  widest  at  its  most  efficient  point.  This  will  not  of  course 
necessarily  give  the  most  efficient  form  of  blade  outline,  but 
the  efficiency  of  such  a  type  of  blade  will  be  greater  than  that 
of  one  possessing  the  same  characteristics  in  other  directions 
but  having  a  uniform  blade  width  throughout. 

If  we  turn  to  the  "  efficiency  curve  "  already  given,  we  shall 
notice  the  following. 

(1)  As  (x)  increases  from  zero  up  to  about  (J)  the  length 
of  the  blade,  the  efficiency  rises  very  rapidly. 


30  AIR-SCREWS 

(2)  As  (x)  increases,  from  the  value  corresponding  to  the 
point  of  maximum  efficiency,  to  its  maximum  value  (i.e.  the 
length  of  the  blade),  the  efficiency  steadily  decreases,  although 
somewhat  slowly  compared  with  its  initial  rise  from  zero  up 
to  its  maximum  value. 

(3)  Hence,  after  the  point  of  maximum  efficiency  has  been 
reached,  the  blade  becomes  less  and  less  efficient  as  we  proceed 
to  the  tip.     Hence  the  useful  work  done  by  the  blade  elements 
decreases  towards  the  blade  tip. 

(4)  This  at  once  suggests  the  advisability  of  making  the 
actual  blade  widths  proportionally  less  at  the  outside  radii  than 
they  would  be  if  made  exactly  proportional  to  the  efficiency 
curve.      The  second  curve  in  Fig.  (9)  would  then  be  a  more 


efficient  shape  than  the  efficiency  curve.  The  maximum  ordinate 
is  now  further  to  the  right  than  in  the  efficiency  curve. 

There  are  however  other  considerations  bearing  upon  the 
problem  of  the  most  efficient  blade  outline  besides  those 
already  given. 

If  we  pursue  still  further  the  analogy  with  an  aeroplane 
wing,  it  at  once  becomes  apparent  that  there  may  be  a  limit  to 
the  useful  blade  width  possible  in  an  air-screw. 

In  the  case  of  an  aeroplane  having  two  or  more  superposed 
surfaces  (e.g.  as  in  a  biplane)  it  is  well  known  that  the  lift  of 
the  lower  wing  is  affected  quite  appreciably  by  the  "  wash  "  of  the 
top  wing,  and  that  the  smaller  the  vertical  distance  between 
the  two  surfaces  the  greater  is  the  loss  in  lift  on  the  lower  wing. 

This  vertical  distance  between  the  wings  of  an  aeroplane  is 


BLADE  SHAPE  AND  EFFICIENCY 


31 


known  as  the  "  gap,"  and  the  ratio  between  this  distance  and 
the  width  of   each  wing    in    the   line  of  flight   is  known   as 

the  "  |f     ,  "  ratio.     This  ratio  has  usually  a  value  of  between 
chord 

(•8)  and  (1*2)  and  is  often  equal  to  unity  in  standard  types  of 
aeroplanes. 

The  value  of  this  vertical  distance  or  "  gap  "  in  the  case  of 
the  blades  of  an  air-screw  is  equal  to 

P.  cos  A 


for  an  air-screw  having  (N)  blades. 


(fl 


FIG.  10. 
It  is  the  value  of  the  vertical  distance  between  any  two 

consecutive  helicoidal  paths  after  ^th  of  a  revolution  and  after 

2 

^ths  of  a  revolution  respectively. 

This  may  be  illustrated  if  we  have  recourse  once  again  to 
the  paper  cylinder. 

Consider  an  air-screw  having  (4)  blades.  And  draw  on  the 
rectangular  piece  of  paper  (7)  parallel  lines  at  equal  distances 
apart,  Fig.  (10). 


32 


AIK-SCREWS 


Now  fold  the  paper  into  a  cylinder  having  a  depth  as  before 
of  (P),  Fig.  (11).  Then  the  lines  so  drawn  will  represent  the 
helicoidal  paths  traced  out  by  the  same  point  on  each  of  the  (4) 
blades  of  the  air-screw.  We  notice  that  the  vertical  distance 
or  "  gap  "  between  any  two  such  consecutive  paths  is  equal  to 

P.  cos  A 


If  then  we  make  the  width  of  the  air-screw  blade  at  any 


FIG.  11. 


radius  (x)  proportional  to  the  vertical  distance  between  any 
two  consecutive  helicoidal  paths,  we  introduce  an  allowance  for 
possible  blade  interference,  Fig.  (12). 
So  that  we  have 

P.  cos  A 


whence 


I  a 


P.  cos  A 

"IT" 


BLADE  SHAPE  AND  EFFICIENCY 
CL 


33 


34 


AIR-SCEEWS 


An  air-screw  having  this  type  of  blade  outline  has  been 
called  by  Mr.  A.  E.  Low  the  "  Eational "  blade.  We  shall 
refer  to  this  type  of  blade  by  that  name  in  what  follows. 
There  are  other  types  of  air-screw  blades,  and  we  shall  now 
consider  two  of  these  further  types. 

The  first  type  considered  is  one  which  would  at  first  sight 
seem  to  be  the  simplest  possible  type  of  blade,  the  blade  of 
uniform  width.  Here  obviously  (&)  is  independent  of  (x), 
since  it  is  the  same  for  all  values  of  the  radii. 


FIG.  13. 
Hence  (&)  =  constant  =  c.f(x),  whence  /(,>;)  =  1,  and  therefore 

(*)=(«). 

And  (c)  has  always  the  value  of  the  ratio 

actual  blade  width  at  radius  (x) 
scale  blade  width  at  radius  (x)  ' 

In  this  case  we  can  take  the  scale  blade  width  as  being 
equal  to  unity. 

A  blade  having  a  constant  chord  width  has  been  called  by 
Mr.  A.  E.  Low  the  "  Normale  "  blade.  We  shall  use  this  name 
when  referring  to  this  type  of  air-screw  blade. 

The  second  form  of  blade  considered  is  one  which  from  a 
constructional  point  of  view  forms  the  limiting  curve  of 
construction  in  which  all  the  lamina  pass  through  the  boss. 

If  in  any  type  of  air- screw  the  blade  widths  at  any  point 
come  outside  this  curve,  it  is  not  possible  to  construct  such  a 
blade  without  using  offsets  on  the  laminae. 

Such  a  form  of  blade  is  shown  in  Fig.  (13). 


BLADE  SHAPE  AND  EFFICIENCY  35 

The  form  of  blade  outline  is  seen  to  be  given  by 

b  =  B.  cosec  (A  +  a), 
whence 

c.f(x)  =  B.  cosec  (A  +  a), 
and  therefore 

c  =  B,     and    f(x) 
and  therefore 

"  PTcos  a  +  2.Tr.x.  sin  a 
and  this  expression  does  not  greatly  differ  from 


the  one  obtained  by  making  (a)  =  0.  This,  as  will  be  seen  later, 
greatly  simplifies  the  subsequent  evaluation  of  the  quantities 
characteristic  of  such  a  type  of  air-screw  blade. 

This  form  of  blade  may  be  termed  the  "  Constructional 
Limit "  type. 

We  shall  now  apply  the  formulae  already  deduced  for 
any  type  of  air-screw  blade  to  the  four  blade  types  already 
considered,  namely : — 

(1)  The  " Efficiency  Curve"  type  of  blade  outline. 

(2)  The  "  Bational"  type  of  blade  outline. 

(3)  The  "  Normale  "  type  of  blade  outline. 

(4)  The  "  Constructional  Limit "  type  of  blade  outline. 

The  three  expressions  for  Thrust,  Torque,  and  Efficiency 
have  been  shown  to  be  given  by 


T  =  p.n2.  Icy.l.  (2.ir,oj-P.  tan  7).  \/P2  +  4.7r2.a32.  dx9 
Jr0 

which  may  be  written  as 

T  =  p,n*.c.  \f(x)4(x).  (2.7r,9;-P.^(t,')  ).  \/F 

Jr0 

giving  the  Thrust  exerted  by  each  blade  of  the  air-screw. 

D  2 


36  AIR-SCREWS 

The  Torque  on  each  blade  is  given  by 

M  =  p.n2.  \Cy.l.x.  (P  +  2.1T.B.  tan  7).  \/P2  +  4.7r2.£2.  dx, 
Jr0 

which  may  be  written  as 
M  =  p.n\c.  \f(x).  4>(x).  x,  (P  +  2.ir.x.  ^(x)  ).  \/P2+4.?r2^  dx. 

Jr0 

The  total  Efficiency  of  each  blade,  and  therefore  of  the  whole 
air-screw,  is  given  by 


.    \cy.b.  (2.7T.E- 


P.  tan  7).  \/P2  +  4.7r2,-e2.  dx 


2.7T.  \cy.b.x.  (P  +  2.7r.£.  tan  7).  \/P 

J^o 

which  may  be  written  as 

P.      /(#).  ^(a?).  (2.ir.a;  -  P.  ^0)  )- 


2.7T.    /(a;), 


and  we  shall  now  apply  these  general  results  to  the  four  types 
of  blade  outline  considered. 


"Efficiency  Curve"  type  of  blade  outline. 

//  \         P.  (2.-n-.a?-P.^(aQ) 
re  J('     ~  2.7r.x.  (P  +  2.7r.aj;  ^(x)  )' 

so  that  the  expression  for  the  Thrust  on  each  blade  becomes 


•r 


...  .  \/P2-f-4.7r2^2.  (2.ir.a;-P.  ^))2.  dx 

2.7T 


BLADE  SHAPE  AND  EFFICIENCY  37 

The  Torque  on  each  blade  is  given  by 

M 


=  £^?.  fj(a>).  (2.7r,*-P. 

J/7r        >o 


The  total  Efficiency  of  the  whole  blade  is  given  by 

rr 

I   (f)(x).    (2.7T.OJ-P.   ^0))! 


2.7T.  \<t>(x).  (2.7r.#  — P.  tfrfa)).  vP2  +  4.7r'2.a52.  dx 
Rational "  type  of  blade  outline. 


N. 

so  that  the  expression  for  the  Thrust  on  each  blade  becomes 

2     Po^2c    fr 
T  =  -— -'— i—  '.  \x.  6(x).  (2.TT.X  —  P.  ty(x)\  dx. 

N  Jr. 

The  Torque  on  each  blade  of  the  air-screw  is  given  by 

Jro 

And  the  total  Efficiency  of  the  whole  blade  is  given  by 
P.    lx.<t>(x).  (2. TT.X-  P.  ^(x)).dx 


2.7T.    <x?.  $(x).  (P4-2.ir.ic.  ^(x)).  dx 


It  will  be  noticed  that  in  the  case  of  these  expressions  for 
the  "Kational"  blade,  the  making  of  the  blade  width  at  any 
radius  proportional  to  the  "  gap  "  has  led  to  a  great  simplification 
being  introduced  into  the  results  of  the  analysis. 


38  AIR-SCREWS 

This  becomes  at  once  apparent  when  we  consider  the 
particular  case  of  uniform  section  over  the  whole  blade. 

</>(V)  and  ^r(x)  are  then  constants,  and  the  three  expressions 
given  above  can  very  easily  be  evaluated. 

In  this  case  the  value  of  (c),  which  is  the  value  of  the 

"  -         '   ratio  of  the  blades,   is   usually  chosen   beforehand, 
gap 

and  a  good  value  for  this  constant  appears  to  be  about  one- 
third  to  one-quarter.     Mr.  A.  R.  Low  suggests  the  rule : 

1  <  m  <  4,  where  (m)  is  the  reciprocal  of  (c). 

If  (c)  be  evaluated  from  outside  considerations  of  B.H.l*. 
available,  etc.,  and  therefore  from  the  formula  given  for  the 
Torque,  it  is  possible  to  see  whether  there  is  likely  to  be 
any  appreciable  interfering  action  between  the  blades  due 
to  forms  of  blade  sections  and  therefore  necessary  blade 
widths  used,  etc. 


"  Normale  "  type  of  blade  outline. 

Here  (b)  =  (c),  and  f(x)  =  1,  so  that   the  Thrust  on  cacti 
blade  is  given  by 


T  =  p.n*.c. 
The  Torque  on  each  blade  is  given  by 


_  _ 
M  =  p.tf.c.      x.  <£(>>).  (P  +  2.7r.#.  ^-(a?)).  \/P2  +  4.7r2.^.  dx. 

Jr0 

And  the  total  Efficiency  of  the  whole  blade  is  given  by 


V  = 


„/>, 

Jr0 


.TT2.^.  dx 


BLADE  SHAPE  AND  EFFICIENCY  39 

"  Constructional  Limit"  type  of  blade  outline. 

•\/PM-~4  7T2  X1 

Here /(a?)  =  p  '     — ,  so  that  the  Thrust  on  each  blade 

is  given  by 


The  Torque  on  each  blade  is  given  by 
M  =  P 


And  the  total  Efficiency  of  the  whole  blade  is  given  by 


'    \     fr\ 

Jr.' 

As  can  be  seen  from  the  above  three  expressions  for  Thrust, 
Torque,  and  Efficiency,  the  evaluation  of  each  expression  is  a 
simple  matter  if  we  take  <$>(%)  and  ty(x)  as  being  constants, 
that  is  if  we  assume  the  air-screw  to  have  a  uniform  section 
throughout.  The  approximation  thus  introduced  does  not  affect 
the  accuracy  of  the  expressions  to  such  a  material  extent  as 
might  be  expected.  The  main  difficulty  in  the  prediction  of 
the  performance  of  an  air-screw  lies  in  the  shape  of  the  blade 
outline  not  being  as  a  rule  readily  capable  of  being  expressed 
by  some  simple  function  of  the  radius  (x). 

We  shall  now  determine  a  few  values  for  the  expressions 
already  deduced  for  Thrust,  Torque,  and  Efficiency,  in  the  four 
cases  already  considered,  when  the  blade  section  is  assumed  to 
be  uniform  throughout.  This  makes  cf>(x)  and  ty(x)  constants, 
and  we  shall  refer  to  these  functions  when  considered  as  being 
constants  by  (cy)  and  (tan  7)  respectively. 

We  shall  assume  that  (r0)  =  0. 

For  the  present  we  shall  confine  our  attention  to  the 
expressions  for  the  efficiency  of  two  only  of  the  types  of  blade 
considered. 


40  AIR-SCREWS 

"  Rational "  type  of  blade  outline. 
We  have 

Cr 

P.  \x.  (2.7T.X  —  P.  tan  7).  dx 

Jr0 =     P.  (4.7r.r-?).P.  tan  7) 

97  =  f~  -  /.  -n  ,  Qirrr  tan  ^ 


!.7T.    p. 
J'o 


And  now,  if  we  write  (Z)  equal  to  the  ratio  of  the  effective 

p 
pitch  to  the  diameter  of  the  air-screw,  i.e.  -r,  we  obtain 


=  2.Z.  (2.7T-3.  tan7.Z) 
77  =      TT.  (4.Z  +  3.7T.  tan  7)  * 

The  substitution  of  (Z)  for  the  ratio  of  pitch  to  diameter, 
p 
-y,  puts  the  expression  into  a  more  convenient  form. 

Mr.  A.  K.  Low  has  deduced  practically  the  same  expression 
for  the  efficiency  of  this  type  of  blade,  his  formula  being 

_         ' 


2.w 

V   ~    o  -\irin 


where  his  (w)  is  equal  to  ^  (as  used  here),  and  his  (M')  and 

Li 

(M'"),  which  are  expressions  for  the  average  values  of  (tan  7) 
over  the  blade,  are  equal  to  (tan  7),  supposed  to  be  constant 
over  the  whole  blade  (as  used  here). 

If  we  make  these  substitutions,  the  two  expressions  for  the 
efficiency  of  this  type  of  blade  become  identical.* 

We  can  introduce  a  still  further  simplification  into  the 
"Rational"  blade  efficiency  formula  by  taking  the  value  of 
(tan  7)  as  1/12.  This  is  a  very  fair  average  value  for  the 
usual  type  of  blade  section  employed  in  practice. 

*  See  Paper  on  "  Air- screws  "  read  before  the  Aeronautical  Society 
in  April,  1913. 


BLADE  SHAPE  AND  EFFICIENCY  41 

We  have  then 

_  2.Z.(8w-Z) 

77    ~~    7T.  (16.Z+7T)' 

This  curve  may  now  be  plotted  against  values  of  (Z).     Such 
a  curve  is  shown  in  Eig.  (14). 


FIG.  14. 

Constructional  Limit "  type  of  blade  outline. 

We  have 


??  = 


P.  f(2.7r^-P.  tan  7)  (P2  +  4.7r2.^2).  dx 


2.7T.     x.(P  +  2.ir.x.  tan 

Jr0 

5.P.(6.7r.P2.^-12.P3. 


.7r'2^2.  dx 


.^2.  tan  7) 


.P2.^.  tan 


.7r3.rf3.  tan  7)' 


42  AIE-SCEEWS 


and  now,  as  before,  making  the  substitution  (Z)  =  -y,  we  obtain 

Cv 

5.Z.  (G.T^Z^J^.Z^an  7-f3.7r3-4.772.Z.  tan7) 
77  ~  TT.    30.Z3  +  20^rrZ2.  tan 


And  if  we  make  (tan  7)  =  1/12,  this  reduces  to 
5.Z.  (18.7r.Z2-3.Z3  +  9.7r3-7T2.Z) 

=     7T.   (90.Z3  +  5.7T.Z-+  45.77  2.Z  +  3.7T3)' 

and  this  curve  may  likewise  be  plotted  against  (Z).  The  curve 
of  this  function  is  shown  in  Fig.  (14). 

It  will  be  noticed  from  the  two  formulae  deduced  for  both 
the  "  Eational  "  and  "  Constructional  Limit  "  efficiencies,  that 
neither  expression  contains  (d),  the  diameter  of  the  air-screw. 

This  is  as  might  be  expected,  since  (ij)  is  non-dimensional, 
and  hence  the  expression  for  the  same  should  only  involve 
ratios  such  as  (Z). 

If  we  make  (tan  7)  equal  constant,  and  therefore  assume 
that  (cy)  is  constant  over  the  blade,  we  may  obtain  quantitative 
expressions  immediately  evaluable  for  the  case  considered  of 
the  "  Eational  "  blade  shape.  These  are  then  as  follows  :— 

-d2.  (2.7r.d  —  3.  P.  tan  7) 
"12 


•\r  M  —  c-Tf-P-Y-^-Cy-n2.  (4.P  +  3.7T.  tan  7.  d) 
48 

,         .  2.C.P.7T.^  ,        .  . 

and,  since  (b)  =  =,   we   may   obtain   the   above 

N.\/P2-f4.7r2.a;2 

expressions  for  the  Thrust  and  Torque  in  terms  of  the  width  of 
the   air-screw  blade  at  the  tip,  i.e.  when  (x)  is  equal  to  (?•). 
We  may  denote  this  tip  blade  width  by  (br). 
These  expressions  then  become 


_  5r.9ia.p.^VFT7r'5Q2.'  (2.7r.^-3.P.  tan  7) 
12 


.n'2.d2.  (4.P-f  3.?r.  tan  7.  <j). 


BLADE  SHAPE  AND  EFFICIENCY  43 

and  the  value  of  (£r),  the  width  of  the  air- screw  blade  at  the 
tip,  is  given  by 

C.P.TT.d 


In  deciding  upon  the  best  form  of  blade  outline  for  the 
air-screw, -an  ideal  condition  would  appear  to  be  obtained  when 
the  velocity  in  the  slip  stream  is  everywhere  parallel  to  the 
axis  and  uniform,  that  is  when  the  thrust  at  any  radius  is 
proportional  to  that  radius.  The  ideal  thrust  grading  diagram 
would  thus  be  a  straight  line  passing  through  the  origin  at  the 
boss  centre. 

We  can  investigate  the  form  of  blade  outline  necessary  to 
secure  such  a  condition  from  the  results  already  obtained  in 
the  general  case. 

It  has  already  been  shown  that  for  any  form  of  blade 
outline  whatever,  the  thrust  at  any  radius  is  given  by 


(IT  =  c.M2xx.2.7r.x-?.<>rx.z+4:.'Tr*.x2.  dx 


and  the  condition  specified  is  that 

dT  =  m.x.dx, 
whence 

m.x  =  c.pMz.f(x)4(x).(2.Tr.x^f. 
and  therefore 


__ 
c.p.n*.<t>(x).(2.7r.x  -  P.^(a;)).  v/P2  -f  4.7r2.^' 

which  gives  the  necessary  form  of  blade  outline  to  satisfy  these 
conditions. 

Whence  it  is  only  necessary  to  plot  a  curve  of 


to  obtain  the  required  blade  outline  for  any  specified  outside 
conditions. 


44  AIR  SCREWS 

It  will  be  noticed  that  when  (#)  has  the  value  of 


that  is  at  a  distance  of  about  one  inch  from  the  boss  centre,  the 
blade  outline  curve  f(x)  becomes  discontinuous,  being  of  the 
form  shown  in  Fig.  (14A). 

Thus  in  estimating   the  characteristics  of  such  a  type  of 


FIG.  14A. 

blade  it  is  impossible  to  integrate  from  the  value  (r0)  =  0,  since 
the  curve  becomes  discontinuous  at  the  value    -J^       of  the 

2.7T 

radius. 

It    becomes    necessary    therefore,    when     evaluating    the 
expressions  characteristic  of  such  a  type  of  blade,  to  take  for 


BLADE  SHAPE  AND  EFFICIENCY  45 

the  value  of  (r0)  a  value  greater  than  _i^i?2.     Since  this  value 

2.7T 

is  usually  very  small,  it  will  be  sufficient  to  take  for  (r0)  a 
value  of  anything  from  say  (-^Q)  to  (J)  of  the  length  of 
each  blade. 

It  is  interesting  to  see  how  the  efficiency  of  this  type  of 
blade  compares  with  those  types  already  investigated. 

The  efficiency  of  any  type  of  blade  is  given  by 


(r  ________ 

P.     f(x).  $(x).  (2.7r.£-P.  Tjr(#))VP24-4.7ra.sc'2.  dx 

=  Jr0  ' 

r          2     -  v/  ; 
.j 


.7r2.ic2.  dx 


and  taking  the  value  of 


,,  s  _ 
~ 
we  obtain 


Tr 

L")          I       /yi       //•O'* 
L.It//.     C6tXy 

Jn 


p 

and  this  becomes,  after  making  the  substitutions      =  (Z),  and 

Cb 

(r0)  =  -Q,  and  ^(x)  =  tan  7  =  constant, 

=   _  3.772.Z 


and  if  tan  7  =  — , 
12 


" 


.  log, 


46  AIR-SCREWS 

It  would  appear  that  the  curve  of  efficiency  plotted  against 
values  of  (Z)  for  this  type  of  blade  does  not  greatly  differ  from 
those  already  given.  The  efficiency  for  the  respective  values 
of  (Z)  is  slightly  higher  than  that  obtained  for  the  "  Rational " 
blade  under  similar  conditions  of  blade  section  and  angles 
of  attack. 


47 


CHAPTER   IV. 

BLADE    SECTIONS,    AND   WORKING    FORMUL/E. 

IN  air-screws  manufactured  for  and  used  upon  existing  types 
of  air-craft  it  will  almost  invariably  be  found  that  the  blade 
section  changes  from  the  tip  to  the  boss,  and  this  is  in  fact  a 
necessity,  from  considerations  of  strength  due  to  the  stresses  in 
the  blades  caused  by  the  thrust  exerted  by  the  air-screw  and 
the  centrifugal  action  due  to  the  rotation  of  the  whole  air-screw 
about  its  axis  at  the  boss  centre. 

Aerodynamically  the  use  of  a  varying  form  of  section  along 
the  blade  usually  somewhat  tends  to  decrease  the  efficiency  of 
the  air-screw  as  a  whole,  although  this  probably  does  not 
amount  to  much.  That  is  to  say  that  the  efficiency  of  any 
type  of  air-screw  blade  having  a  section  varying  from  the  boss 
to  the  tip  is  less  than  that  of  a  blade  of  similar  outline  but 
having  a  uniform  section  throughout,  provided  that  such  a 
section  is  of  such  a  form  as  to  have  a  value  of  (tan  7)  smaller 
than  the  average  value  of  (tan  7)  on  the  other  blade,  and 
equal  to  the  smallest  value  of  (tan  7)  on  any  section  there 
employed. 

As  a  rule,  for   sections   of   similar   shape,  the  thinner  the 

fil  1  f*  1C  Tl  f*^S 

section,  i.e.  the  less  the  value  of  the  - — r—  -,—  ratio,  the  lower 

the  value  of  (tan  7),  and  hence  the  greater  the  overall  efficiency 
of  the  whole  blade  employing  such  a  section. 

This   only   applies,  however,  between   certain   fairly   well 

,  ,.      thickness      ,. 
defined  limits,  for,  after  a  certain  value  or  the  — g — -j—  ratio 

is  reached,  the  section  becomes  less  and  less  efficient,  that  is  has 
a  larger  and  larger  value  of  (tan  7),  until  the  limit  is  reached, 


48  A  IE-SCREWS 

when  the  "  camber  "  of  the  section  altogether  disappears  and  the 
section  ultimately  becomes  a  flat  plate  for  which  the  value  of 

the  YJ —  ratio  is  quite  small,  being  of  the  order  of  (*7)  :  (1)  at 
.Drag 

the  most  efficient  angle  of  incidence. 

As  a  rule,  in  designing   an  air-screw  to  fulfil   any  given 
outside  conditions,  the  forms  of  the  blade  sections  are  chosen 


ChoYd      Angle     of    Incidence    (ctegrte*) 

FIG.  15. 

so  as  to  effect  as  far  as  possible  a  compromise  between 
considerations  of  aerodynamical  efficiency  and  the  necessary 
strength  of  the  various  portions  of  the  blade. 

The  aerodynamical  considerations  of  overall  blade  efficiency 
have,  however,  already  been  dealt  with,  and  the  expression  for 
the  efficiency  of  any  type  of  blade  shape  put  into  a  suitable 
form  for  purposes  of  air-screw  design,  so  that  we  may  proceed 


BLADE  SECTIONS,  AND  WOEKING  FORMULAE    49 

to  select  our  sections  along  the  blade  without  stopping  to 
consider  whether  it  will  be  possible  to  express  analytically 
the  characteristics  of  the  same  in  estimating  the  Thrust,  Torque, 
and  Efficiency  of  the  whole  blade,  providing  of  course  that  the 
forms  of  the  blade  sections  chosen  are  such  that  they  conform 
to  sections  of  which  the  characteristics  are  known  from  tests 


Chord  A  ng/e   of  Incidence  fdtyrreij 
FlG.   16. 

carried  out  in  a  wind-channel  for  the  same  when  considered 
as  aerofoils  moving  in  a  straight  line. 

A  somewhat  typical  series  of  such  sections  are  given  in  the 
(<  Technical  Report  of  the  Advisory  Committee  for  Aeronautics 
for  1911-1912,"  and  their  characteristics  are  shown  plotted  in 
Figs.  (15),  (16),  (17),  (18),  (19),  (20),  (21). 

It  will  be  noticed  that  the  maximum  value  of  the  ^p- 

Drag 


50 


AIR-SCKEWS 


ratio,  and  hence  the  minimum  value  of  (tan  7),  occurs  in  all  the 
sections  at  an  angle  of  approximately  (4)  degrees. 

We  may  of  course  use  any  " angle  of  attack"  we  please  for 
each  section  along  the  blade,  but  it  is  advisable  to  use  the  angle 
corresponding  to  the  least  value  of  (tan  7)  fur  each  section 
considered,  as  this  gives  a  better  overall  efficiency  for  the  whole 
blade,  since  the  efficiency  of  any  section  is  equal  to 

tan  A 
tan  (A +7)' 


Chord    Angle    of   Inc.  i  d  e.  nee  • 


FIG.    17. 


and  hence  the  smajler  the  value  of  (tan  7),  and  hence  of  (7), 
the  greater  the  efficiency  at  that  section. 

We  now  proceed  then  to  a  consideration  of  the  design  of 
any  type  of  air-screw  to  fulfil  any  given  specified  set  of 
conditions. 


BLADE  SECTIONS,  AND  WORKING  FORMULAE    51 

It  is  usual   to   start  with   the   following  data,  which  are 
supposed  to  be  fixed  from  outside  considerations. 

(1)  (V)          The  horizontal  velocity  of  the  air-craft. 

(2)  (?i)          The  speed  of  revolution  of  the  air-screw  in 

revs. /second. 


Chord    Anyle    of  Incidence. 

FIG.  18. 


(3)  (d)          The   diameter  of  the  air-screw.      This  is 

usually  fixed  from  considerations  of 
ground  clearance,  etc.,  and  is  usually 
made  as  large  as  the  design  of  the  air- 
craft will  permit.  This  seems  to  be 
fairly  standard  practice  at  present. 
Experimental  research  is  required  before 
this  point  can  be  definitely  settled. 

E  2 


52  AIR-SCEEWS 

(4)         <p(x)         These  functions  of  (%)  and  (tan  7)  depend 

and  upon  the  form  of  the  sections  employed 

•fy(x)  at  various  radii  along  the  blade  and  their 

respective  "  angles  of  attack,"  and  may 

be  plotted  when  the  form  and  position 

of  these  relative  to  the  boss  centre  are 

known. 


Chc-rd    Anyle   of  Incide  nc  e  Idc y 

FIG.  19. 


(5)  •  (H)  The  B.H.P.  of  the  motor  multiplied  by 
the  efficiency  of  the  transmission  (if 
any)  supplied  to  each  air-screw  (if  there 
are  more  than  one). 

We  now  proceed  to  draw  out  the  blade  shape  of  the  air- 
screw, making  ordinates  on  the  curve  represent  proportional 


BLADE  SECTIONS,  AND  WORKING  FORMULA    53 

widths.  Here  experience  and  a  study  of  the  particular  type  of 
air-craft  to  which  the  air-screw  is  to  be  fitted  will  guide  us. 

Portions  of  the  air-craft  coming  within  the  air-screw's  disc 
of  revolution  must  be  taken  into  account  when  designing  the 
blade  shape. 

An  air-screw  fitted  to  a  rotary  motor  having  a  large  cowl, 
for  instance,  will  not  probably  require  so  large  a  blade  width 


Chord  Anyle    of  Inc  i de  nc  t 

FIG.  20. 


for  best  efficiency  at  the  inside  radii  near  the  boss  as  would 
otherwise  be  advisable  if  the  air-screw  were  working  free  from 
any  such  obstructions  in  the  air-flow. 

In  the  theory  of  air-screw  design  as  set  forth  in  this  book 
no  quantitative  notice  is  taken  of  such  stationary  parts  of  the 
air-craft,  and  to  do  so  would  be  in  most  cases  an  exceedingly 
difficult  problem. 


54 


AIR-SCKEWS 


A  study  of  the  efficiency  curve  for  the  blade  will  also  help 
us  in  the  designing  of  a  suitable  blade  outline. 

Bearing  then  these  considerations  in  mind  we  commence  the 
designing  of  our  air-screw  as  follows  : — 

(1)  Plot  the  efficiency  curve 
P.    2/ 


J    Incidence    (deyrees) 


FIG.  21. 

We  do  not  as  yet  know  the  value  of  the  function  ^r(,r), 
but  for  a  first  approximation  we  may  take  this  as 

being  constant  and  equal  to  say  T^. 

±Z 

In  slow  running  air-screws  this  curve  of  blade  efficiency 
will  be  found  to  be  almost  a  parallel  line  with  the 
(x)  axis,  except  close  up  to  the  boss,  where  it  runs 
down  to  the  origin  very  rapidly. 


BLADE  SECTIONS,  AND  WORKING  FORMULA    55 

In  high  speed  air-screws  the  change  in  slope  of  the 
efficiency  curve  will  be  more  marked,  and  this 
brings  us  to  a  consideration  of  the  best  blade  shape 
under  these  conditions. 

As  an  example  of  a  good  blade  outline  for  an  air-screw,  the 
four-bladed  air-screw  as  used  on  the  Royal  Aircraft  Factory 
aeroplanes  may  be  cited.  This  screw  revolves  at  a  speed  of 
900  r.p.m.,  which  is  if  anything  rather  on  the  slow  side. 

We  may  however  take  it  as  a  fairly  good  general  rule  that 
in  designing  an  air-screw  blade,  the  maximum  ordinate  on  the 
blade  outline  curve,  that  is  the  point  where  the  blade  is  widest, 
should  be  somewhat  nearer  the  tip  of  the  blade  than  the  point 
of  maximum  efficiency  as  given  by  the  efficiency  curve.* 


FIG.  22. 

It  is  not  proposed  here  to  discuss  the  theoretically  best  form 
for  the  blade  outline  function  f(x),  as  to  do  so  would  be  a  very 
difficult  matter  when  treated  generally.!  Experience  is  a  good 
guide  in  choosing  a  suitable  shape  for  the  form  of  this  function. 

Taking  then  the  blade  outline  as  being  something  of  the 

*  Recent  tests  on  aerofoils  with  elliptical-shaped  ends  have  shown  a 
marked  improvement,  in  —  ratio,  over  similar  aerofoils  with  square- 
cut  ends,  provided  that  the  section  of  blade  in  the  former  case  is 
everywhere  geometrically  similar.  This  indicates  the  superiority,  from 
efficiency  point  of  view,  of  blades  with  tapered  tips,  apart  from  any  other 
considerations. 

f  A  general  treatment  of  the  variation  in  (77)  due  to  variation  of  /(#), 
and  the  determination  of  the  form  of  f(x)  giving  to  (77)  a  maximum 
value,  would  require  the  application  of  the  Calculus  of  Variations,  and 
is  beyond  the  scope  of  the  present  work. 


56  AIE-SCKEWS 

form  shown  in  Fig.  (22),  we  may  proceed  to  the  determination 
of  the  complete  design  of  the  air-screw. 

/o\  T±   •     ^  fi     ^     thickness      ,.        „  , . 

(2)  It  is  first  necessary  to  fix  the  — r — T—  ratios  of  the 

sections  along  the  blade,  and  here  again  practical  experience 
will  help  us.  It  is  undesirable  to  make  the  blade  sections  too 
thin,  especially  near  the  boss,  as  this  leads  to  undue  flexibility 
in  the  blade,  with  corresponding  losses  in  efficiency. 

In  an  air-screw  having  a  diameter  of  (8)  or  (9)  feet,  the 
following  proportioning  of  the  sections  along  the  blade  would 
seem  to  be  fairly  standard  practice  :— 

At  a  radius  of  (1)  foot  from  the  boss  centre, 

Section  No.  7,  Fig.  (21). 

(2)  feet         „  „         „         „  6,  Fig.  (20). 

(3)  „  „  „         „         „  5,Fig.(19). 

(4)  „'          „  „         „         „  4  Fig.  (18). 

Having  then  selected  a  suitable  series  of  aerofoil  sections 
and  having  spaced  them  along  the  blade,  we  may  proceed 
to  the  determination  of  the  two  functions  </>(#)  and  ^(#) 
characteristic  of  such  sections. 

The  respective  "  angles  of  attack  "  of  the  various  sections 
employed  will  of  course  be  the  angles  corresponding  to  the 
least  value  of  (tan  7)  in  each  case,  since  this  will  give  the 
highest  efficiency  for  any  specified  spacing  of  the  sections  and 
form  of  blade  outline. 

In  the  case  considered,  these  "  angles  of  attack  "  are  taken 
as  being  constant  and  equal  to  (4)  degrees,  since,  as  already 
stated,  and  as  a  reference  to  the  curves  given  in  Figs.  (18, 
19,  20,  21)  will  show,  the  respective  minimum  values  of  (tan  7) 
occur  approximately  at  the  same  angle  of  incidence  in  each 
case,  namely  (4)  degrees. 

In  this  connection  it  does  not  seem  to  be  unreasonable  to 
assume  that  sections  lying  between  any  two  selected  sections 
along  the  blade  will  have  characteristics  intermediate  between 
those  of  the  two  respective  outside  sections  considered,  and 


BLADE  SECTIONS,  AND  WOEKING  FORMULAE    57 

quantitatively  will  approximate  to  the  values  of  the  character- 
istics shown  on  a  smooth  curve  drawn  between  the  various 
points  along  the  blade  corresponding  to  the  characteristics  of 
the  selected  sections  at  those  points. 

We  shall  then  obtain  a  series  of  points  for  the  values  of 
(cy)  and  (tan  7)  for  the  various  radii  chosen,  and  if  smooth 
curves  be  drawn  through  these  points  we  shall  obtain  the  two 
curves  denoted  by  <£(^)  and  ty(x). 

Fig.  (23)  gives  these  two  curves  for  the  sections  and  spacing 
of  the  sections  considered  and  for  a  uniform  "  angle  of  attack  " 
of  (4)  degrees. 

We  are  now  able  to  read  off  from  the  two  graphs  plotted 
of  <f>(x)  and  -fy(x)  the  respective  values  of  these  two  functions 
for  any  value  of  (x)  considered,  that  is  for  any  point  along  the 
blade. 

(3)  It  now  only  remains  to  determine  the  true  value  of  the 
blade  width  for  every  point  along  the  blade. 

The  efficiency  of  the  whole  blade  can  be  found  at  once,  since 
we  know  the  values  of  the  two  functions  </>(x)  and  ty(x)  for 
each  value  of  (x)  considered. 

The  value  of  (&),  the  true  blade  width  at  any  radius  (#),  can 
be  obtained  in  the  following  manner. 

We  already  know  that 

b  =  c.flx), 

where  f(x)  denotes  the  scale  blade  width  for  each  value  of  (x) 
considered,  and  the  value  of  f(x)  is  thus  the  value  of  the 
ordinate  on  the  curve  of  proportional  blade  widths  already 
drawn.  So  that  we  have  only  to  find  the  value  of  (c)  in  order 
to  be  able  to  completely  determine  the  value  of  (&)  for  any 
value  of  the  radius  (x). 

It  has  already  been  shown  that  the  Torque  (M)  of  each 
blade  of  the  air-screw  may  be  expressed  as 

M  =  c.p.n\  \x.f(x).  $(x).  (P  +  2.7T,*.  f  (^))VPH-"4;7r^;  dx. 

Jr0 

And  further  (N.M.2.7r.w)  is  proportional  to  the  horse-power 


58 


AIE-SCEEWS 


o  * 


'  /       vW 

y     <£X/ 

to 

V 

I    / 

V 

U. 

CO 

o? 

i 

& 

BLADE  SECTIONS,  AND  WOEKING  FORMULAE    59 

available  to  turn  the  whole  air-screw,  where  (N)  denotes  the 
number  of  blades. 

Hence  if  we  use  Ib./ft./sec.  units  we  get  that 

IN  .  ivi.-j.7T.  n       .„ 

^TK =  J~ij 

ooO 

where  (H)  is  the  available  B.H.P.  after  allowing  for  losses  in 
transmission  (if  any  transmission  is  employed). 

We  also  have  the  value  of  (p)  as  being  equal  to  ( •  00238)  in 
Ib./ft./sec.  units. 

So  that  we  can  at  once  write 


=  H 


whence 

550.  H 


(r 

2.7T.nS.N.p.       X.f($).  <£(». 

Jr0 


and  this  gives  the  necessary  value  of  (c)  for  any  given  outside 
conditions. 

It  will  be  noted  that  all  dimensions  must  now  be  measured 
in  feet. 

We  proceed  then  to  the  evaluation  of  the  blade  width 
constant  (c). 

Since  we  have  not  obtained  the  functions  <j>(x)  and  ty(x)  in 
an  algebraic  form,  that  is  we  do  not  know  the  equations  to 
these  two  curves,  we  cannot  evaluate  the  definite  integral 


I, 


x.f(x).  </>(». 


algebraically,   and    hence    must    employ   a  graphical  method 

throughout.     This  also   applies   to  the  evaluation  of  (/;),  the 
efficiency  of  the  whole  blade. 

In    order    then    to    obtain    the   value   of   (c)    it  is    first 


60  .  AIE-SCEEWS 

necessary  to  determine  graphically  the  value  of  the  definite 
integral 

\X.f(x).  $(X).  (P-f2.7T.JE.   ^)).V^  +  ^7T\X\  dx, 

h 

and  we  may  proceed  to  do  this  as  follows. 
Plot  the  graph  of  the  function 
x.  f(x).  $(x] 


to  some  convenient  scale  against  values  of  (x),  the  radius  from 
the  boss  centre  in  feet. 

y 
We  already  know  the  value  of  (P) ;  it  is  —  ;  and  both  (V) 

and  (n)  are  known  to  start  with,  since  they  are  assumed  to  be 
fixed  from  outside  considerations. 

The  value  of  f(x)  for  any  radius  (x)  is  known,  since  it  is  the 
value  of  the  ordinate  on  the  scale  blade  width  already  drawn. 
The  value  of  f(x)  is  to  be  measured  in  feet,  that  is  to  say  we 
must  take  some  convenient  scale  on  the  graph  paper  to  represent 
so  many  inches  equal  to  one  foot. 

The  respective  values  of  </>(x)  and  ^r(x)  are  determined  at 
once  from  a  reference  to  the  two  curves  already  plotted  of  these 
functions  for  any  value  of  (x). 

»  Having  then  taken  a  sufficient  number  of  values  of  (x),  and 
having  determined  the  corresponding  values  of  the  function  the 
graph  of  which  we  are  plotting,  a  smooth  curve  drawn  through 
the  points  so  obtained  will  give  the  curve  required. 

Now  draw  two  ordinates  from  the  (x)  axis  at  the 
points  (r0)  and  (r)  respectively  until  same  cut  the  curve  just 
plotted. 

Then  the  area  of  the  enclosed  figure  so  obtained,  that  is  the 
figure  contained  by  the  curve,  the  two  extreme  ordinates  at  (r0) 
and  (r),  and  the  (x)  axis,  is  the  value  of  the  definite  integral 
required. 

The  areas  of  closed  figures  of  this  kind  are  most  easily 
obtained  by  means  of  a  planimeter. 

It  is  to  be  noted  here  in  this  connection  that  the  actual  area 
of  the  figure  so  obtained  will  be  in,  say,  square  inches,  and 


BLADE  SECTIONS,  AND  WOKKING  FORMULAE    61 

hence  that  it  must  be  multiplied  by  some  constant  in  order  to 
find  the  real  value  of  the  definite  integral. 

The  value  of  this  constant  will  depend,  of  course,  upon  the 
scale  employed  in  the  plotting  of  the  graph. 

Thus,  if  the  ordinate  on  the  curve  corresponding  to  any 
value  of  (x)  has  a  value  obtained  from  the  formula 


T  f(l-\   d\(t>\      (~P  4-  9    TT   V  ^(^\ 

^•J vV'V^V  /•  \     -T^.TT.X. Y v*v 

of  such  and  such  an  amount,  this  value  will  probably  not  be 
able  to  be  represented  on  the  graph  paper  used,  since  it  may  be 
very  much  too  large  when  taken  to  the  same  scale  as  that  of 
the  (x)  axis,  and  hence  the  scale  of  "  heights  "  or  ordinates  will 
probably  have  to  be  made  much  smaller  than  the  scale  used  for 
the  (x)  axis. 

Having  then  obtained  the  value  of  this  area  in  the  required 
dimensions,  we  may  determine  the  value  of  (c)  at  once. 

It  is  given  by 

550.  H 

/>    ^— —     , , 

2.7r,n3.N.p.  (area  of  figure  obtained)' 

where  (p)  has  the  value  of  (-00238)  as  already  given. 

We  have  then  that  the  true  or  necessary  blade  width  at 
each  radius  (x)  is  the  value  of  the  scale  blade  width  at  that 
radius  multiplied  by  the  value  of  the  constant  (c)  already 
found. 

Or 

I  =  cf(x). 

The  value  of  f(x)  for  any  radius  (x)  will,  of  course,  be 
measured  off  the  curve  of  this  function  already  drawn,  and 
its  corresponding  real  value  found  by  reference  to  the  scale 
employed  in  drawing  the  curve. 

Thus,  if  1  inch  on  the  curve  ordinate  represents  1  foot  as 
the  actual  scale  blade  width,  a  value  measured  at  any  radius 
of,  say,  ( '  75)  inch  as  the  ordinate  of  the  curve  at  that  radius 
would  represent  an  actual  scale  blade  width  of  ( •  75)  foot,  that 
is  (9)  inches. 

And  further,  if  the  value  of  (c)  was  found  to  be,  say,  (1  •  2), 
then  the  true  or  actual  blade  width  at  this  point  would  be 


62  AIR-SCREWS 

equal  to  (9  X  1*2)  inches,  that  is  (10*8)  inches.  And  this 
would  be  the  width  of  the  blade  at  that  radius  to  be  used  in 
the  construction  of  the  air-screw. 

We  may  obtain  the  value  of  the  total  efficiency  of  each 
blade,  and  hence  of  the  whole  air-screw,  in  a  similar  manner 
as  follows. 

Plot  the  two  curves 


4.7r^ (1) 

and 


h4.^2 (2), 

which  is  the  one  already  plotted,  and  which  therefore  it  is 
unnecessary  to  replot. 

Take  the  area  of  the  figure  enclosed  by  (1),  the  extreme 
ordinates  at  (r0)  and  (r),  and  the  axis  of  (x). 

The  area  enclosed  by  (2)  has  already  been  obtained  from  the 
evaluation  of  (c). 

Divide  the  area  enclosed  by  (1)  by  the  area  enclosed  by  (2), 

p 
and   multiply   the   result   by  ^ — .     The    answer   will   be   the 

efficiency  of  the  air-screw. 

It  will  not,  of  course,  be  necessary  to  trouble  about  scale 
constants,  etc.,  in  determining  the  value  of  the  efficiency  of  the 
whole  blade,  as,  provided  that  the  two  graphs  (1)  and  (2)  are 
drawn  to  the  same  scale,  it  will  only  be  necessary  to  divide 
their  actual  areas  one  by  the  other  in  whatever  dimensions 
these  two  areas  are  obtained,  provided,  of  course,  that  each  area 
is  measured  in  the  same  dimensions. 

In  the  determination  of  (c),  and  hence  of  the  real  blade 
width  for  each  radius,  it  will  usually  be  found  that  the  true 
blade  widths  for  each  point  along  the  blade  will  be  somewhat 
smaller  than  those  actually  employed  in  practice  on  a  similar 
form  of  air-screw.  This  is  due  to  the  fact  that  the  calculated 
Torque  (M)  is  higher  than  the  actual  Torque  found  in  practice, 
and  hence  that  the  necessary  blade  widths  at  each  radius  will 
have  to  be  larger  than  those  given  by  the  theory. 

Of  course  no  one  for  a  moment  supposes  that  the  theory  of 


BLADE  SECTIONS,  AND  WORKING  FORMULA    63 

air-screw  design  based  on  an  analogy  with  aerofoils  moving  in 
a  straight  line  is  absolutely  exact.  It  is  merely  a  very  useful 
theory,  the  results  of  which  conform  very  closely  with  those 
actually  obtained  by  experiment. 

It  is  difficult  to  estimate  the  amount  of  this  difference 
between  the  calculated  and  actual  Torques,  without  actual 
quantitative  experimental  results,  but  it  would  appear  that  the 

actual  value  of  (c)  is  between  -==-  and  — ..-  times  the  calculated 

/  o  7o 

value  as  given  by  the  theory. 

So  that  after  having  obtained  the  calculated  value  of  (c),  it 
will  be  prudent  to  augment  this  value  by  say  33  %,  that  is 

multiply  the  value  of  (c)  obtained  from  the  formula  by  -=-   . 

7 o 

It  will  be  useful  here  to  have  some  independent  check  upon 
the  working,  so  that  any  arithmetical  slips  in  the  evaluation  of 
quantities  such  as  (c)  may  be  as  far  as  possible  avoided. 

We  may  obtain  a  rather  approximate  check  of  this  kind 
by  reference  to  the  "  Rational "  blade  outline  form  already 
considered. 

If,  in  the  expressions  deduced  for  the  characteristics  of  this 
form  of  blade,  we  assume  that  both  (cy)  and  (tan  7)  are  constant 
over  the  blade,  and  that  (r0)  is  equal  to  zero,  and  further  if 

we   take   the  value  of  (tan   7)  to  be  ^r,  then  we  obtain  the 

\2t 

following  expression  for  the  necessary  blade  width  constant  (c). 

52800.  H 


and  since  as  already  shown 

c  =  — 


where  (br)  denotes    the   tip   blade  width,  we  can   obtain    the 
necessary  value  of  (br)  by  substitution,  thus 

52800.  H 


And  this  then  provides  a  useful  approximate  check  upon 


64  AIR-SCREWS 

the  previous  work  in  the  determination  of  the  true  blade  width 
for  each  radius  (#). 

It  will  be  noticed  from  this  and  previous  expressions  for 
blade  width,  that  doubling  the  number  of  blades  of  an  air-screw 
merely  halves  the  respective  widths  of  same  at  corresponding 
radii. 

This  theory  does  not  in  fact  take  any  notice  of  possible 
improved  efficiency  by  the  use  of  more  than  the  customary  two 
blades  for  an  air-screw.* 

We  may  also  obtain  an  approximate  value  for  the  efficiency 
of  the  whole  blade  by  means  of  the  same  "  Rational "  form  of 
blade  outline. 

This  has  already  been  shown  to  be  given  by 

2.Z.  (2.7T  -  3.  tan  7.  Z) 
TT.  (4.Z  +  3.7T.  tan  7)  ' 

which  reduces  to  the  simpler  form 

2.Z.  (8.7T-Z) 

77    ==    7T.    (7T  +  16.Z) 

when  (tan  7)  has  the  value  of  — . 

Now  there  will  be,  corresponding  to  some  particular  value 
of  (Z),  a  value  of  (rf)  which  will  be  a  maximum,  and  this  will 
therefore  give  the  theoretically  best  value  of  (Z)  and  therefore 
of  (n)  to  use  for  any  given  set  of  conditions,  since  the  values 
of  (V)  and  (d)  will  usually  be  fixed  from  outside  considerations. 

We  may  obtain  the  value  of  this  maximum  efficiency  and 
the  value  of  (Z)  for  which  it  is  a  maximum  as  follows. 

The  condition  for  a  maximum  value  of  (77)  is 

^L-  0 
dZ  ~    U' 

and  this  gives 

Zl  =   ~.  [\/8  +  9.  tan2~r7-3.  tan  7], 

giving  the  requisite  value  of  (Z)  for  a  maximum  value  of  (77). 

*  Except  in  so  far  as  the  greater  the  number  of  blades  employed  the 
higher  the  aspect  ratio  of  each,  and  hence  the  greater  the  efficiency  of 
the  air-screw  due  to  increase  of  aspect  ratio. 


BLADE  SECTIONS,  AND  WORKING  FORMULA    65 


This  expression  is  seen  to  approximate  to  the  value  !^_f 

as  (tan  7)  approaches  zero. 

As  a  rule  this  value  is  in  the  neighbourhood  of  (2)  and 
hence  we  may  say  that,  for  a  near  approximation,  (77)  has  a 
maximum  value  when  (Z)  is  equal  to  (2). 

P        V 

Now  (Z)  =  -    =  -      and  hence  if  (Y)  and  '(d)  are  fixed  by 

Cl>          1l'.(L 

outside  considerations,  the  value  of  (n)  for  which  the  efficiency 
of  the  air-screw  is  a  maximum  is  given  by 

Zl  =  ^d  ~  2' 
whence 

(nL)  =  y-j  revs.  /sec. 

So  that  if 

(V)  =  100  feet/sec., 
(d)  =  10  feet, 

the  speed  at  which  the  air-screw,  or  air-screws,  should  be  run  in 
order  to  obtain  the  maximum  efficiency  would  be  (5)  revs.  /sec., 
that  is  (300)  revs./min. 

This  speed  is  of  course  abnormally  slow  in  the  light  of 
present-day  practice,  although  the  speed  of  revolution  of  the 
air-screws  in  some  of  the  Wright  aeroplanes  is  as  slow  as 
(450)  revs.  /ruin. 

A  curve  of  efficiency  for  values  of  the  ^i  -  ratio  (Z) 

Diameter 

has  already  been  given  for  values  of  (Z)  occurring  in  practice. 

Mr.  H.  Bolas  gives  a  formula  for  the  efficiency  of  an  air-" 
screw  of  good  shape  *  (see  "  Technical  Report  of  the  Advisory 

*  This  formula  is  only  an  approximate  one.     It  is 


"  1  4-  Ic.  tan  y.  cot  6 

-  Z 

Z  4-  7T./t.  tan  y' 

and  if  tan  y  =  — ,  A'  =  *7,  we  get 

Z 


~  Z  +  '188' 
which  may  then  be  plotted  against  (Z). 


66  AIK-SCKEWS 

Committee  for  Aeronautics,  1911-12  "),  and  this  has  also  been 
plotted  against  values  of  (Z).  The  two  curves  of  efficiency  are 
shown  in  Fig.  (14),  and  their  close  general  resemblance  will  be 
noticed. 

It  would  seem  that  at  any  rate  to  a  first  approximation 
the  efficiency  of  almost  any  modern  type  of  air-screw  may  be 
obtained  from  the  formula  or  graph  given  for  the  "  Eational " 
shape. 

As  a  rule,  in  practice  it  is  found  that  the  actual  recorded 
efficiencies  of  air-screws  are  higher  than  those  given  by  the 
theory. 

Since  (Z)  may  be  taken  as  equal  to  (2)  for  a  maximum 
value  of  (??),  the  maximum  value  of  (77)  will  be  obtained  by 
substituting  this  value  for  (Z)  in  the  efficiency  formula.  We 

shall  take  (tan  7)  as  being  equal   to  ^  as  before,  and    then 

we  have  the  value  of  (?;)  as  ('84).  Hence  the  maximum 
overall  efficiency  of  a,n  air-screw  having  a  "  Eational"  blade 

shape  is  (84%)  when  (tan  7)  =  ^  and  (Z)  =  (2). 

Suppose  that  we  wish  to  design  an  air-screw  for  an 
aeroplane  having  a  speed  (Y)  of  (100)  feet/sec.,  a  rotational 
speed  (11)  of  (20)  revs./sec.,  a  diameter  (d)  of  (9)  feet,  and 
having  a  motor  capable  of  developing  an  effective  horse-power 
(H)  of  (100). 

Then 

1°}  and  hence  (P)  -  Y-  =  5  feet. 

(     /Z'    )          • ..  \J  S  Ji> 

(d)  =       9 
(H)  =  100 

Let  the  spacing  of  the  sections  along  the  blade  be  such  as 
already  given  with  a  uniform  "  angle  of  attack  "  of  (4°),  and 
hence  let  the  curves  of  $(#)  and  ^(x)  be  such  as  given  in 
Kg.  (23). 

Suppose  that  the  form  of  /(#),  the  blade  outline,  is  that 
given  in  Fig.  (22). 


BLADE  SECTIONS,  AND  WOEKING  FORMULAE    67 

Then  we  have  to  determine  (1)  The  actual  blade  widths  at 

every    point    along     the 
blade. 

(2)  The  total  efficiency  of  the 
whole  air-screw. 

We  shall  neglect  the  B.H.P.  consumed  in  turning  the 
portion  of  the  air-screw  from  the  boss  centre  to  the  inside 
radius  (r0),  as  by  doing  so  we  shall  be  on  the  safe  side,  since 
our  blade  widths  will  now  tend  to  come  out  larger  than  if  we 
allowed  for  the  B.H.P.  used  near  the  boss.  In  any  case  the 
amount  of  this  B.H.P.  is  very  small. 

We  proceed  to  determine  the  value  of  (c)  in  the  manner 
given  by  plotting  the  graph  of 


x.f(x).  <£(<'-')•  (P- 

taking  values  of  f(z),  <£(#),  and  ty(a')  from  the  curves  already 
plotted  of  these  functions. 

The  graph  is  shown  in  Fig.  (24),  and  the  actual  area  (as 
originally  drawn)  is  (50*25)  sq.  ins.,  between  the  two  extreme 
radii. 

Now  the  scale  of  ordinates  for  this  curve  is  taken  to  be  for 

convenience       of  the  horizontal   scale,  so  that  the  true  area 

o 

enclosed  by  the  curve  will  be  (8  X  50*25)  sq.  ins.,  and,  since 
the  horizontal  scale  employed  is  (2)  ins.  equal  to  one  foot,  this 
area  corresponds  to  (100*5)  sq.  feet. 

Hence,  applying  the  formula  for  (c),  we  get 

550.  H 


2.7r.7i3.N./o.  (Area  of  figure) 

where  (N)  =  (2).  It  will  be  noticed  that  if  we  employ  (4) 
blades  instead  of  (2),  the  value  of  (c),  and  hence  the  value  of 
the  blade  width  at  any  radius,  is  halved. 

If  we  now  multiply  the  value  (2  •  3)  for  (c)  obtained  above 

by  say  -^K ,   so    as    to    allow    for    differences    between    the 

7o 

F  2 


68 


AIR-SCREWS 


calculated  and  actual  Torques,  we   obtain   (3-07)  as   a   more 
exact  value  for  this  constant. 

We  can  now  of  course  obtain  the  true  blade  widths  at  each 
radius  by  multiplying  their  respective  values  as  given  on  the 


£     i 


s« 


5colc    o/   5colr  Blade   Width 
Some     as  Horizontal     5  c  a  I  e  . 


scale  blade   width  curve   by  (3 -07)  when   the   air-screw   has 
(2)  blades. 

We  can  also  check  this  value  by  applying  the  "  Rational  " 
blade  constant  formula  and  this  then  gives 

52800.  H^ 

Tr.n*.p.cy.d?.N.  (16.P  +  TTJ!) V 


BLADE   SECTIONS,  AND  WORKING  FOEMUL^E    69 

=  (-486)  foot,  which  is  the  value  of  the  width  of  the  blade  at 
the  tip,  when  (N)  =  (2).  The  value  of  (cy)  has  been  taken  to 
be  (*36).  This  value  of  the  blade  tip  width  is  about  what 
might  be  expected  for  this  type  of  blade,  where  the  blade 
increases  in  width  towards  the  tip. 

We  may  now  determine  the  value  of  (77),  the  total  efficiency 
of  the  whole  blade.     We  plot  the  curve 


and  take  the  area  of  the  figure  enclosed  by  it,  the  two  extreme 
radii  at  (r0)  and  (?'),  and  the  (,/;)  axis. 

The  area  of  this  curve  as  originally  drawn  is  approximately 
(47  •  6)  sq.  ins.  Eig.  (24). 

The  value  of  the  total  efficiency  of  the  whole  blade  is  then 
equal  to 

47-6  \    /  P  \       /47-6\    /  5  \ 

= 


since  (?)  is  equal  to  (5)  feet. 

The  efficiency  of  an  air-screw  of  this  type  would  therefore 
be  approximately  75  •  3  %. 

The  blade  sections  of  an  air-screw  at  the  inside  radii  near 
the  boss  have  to  be  made  thick  from  considerations  of  strength. 
We  may,  however,  choose  suitable  shapes  for  the  outside  radii 

from  sections  which  have  a  high  value  of  the  ,          ratio  as 

Drag 

aerofoils. 

A  few  typical  examples  of  such  suitable  shapes  are  given 
in  Eigs.  (25),  (26),  (27),  (28).* 

*  See  "  Technical  Eeport  of  the  Advisory  Committee  for  Aeronautics, 
1912-13." 


AIR-SCKEWS 


DCS.CN  /*»e».fo). 


FIG.  25. 


CAorrf  An,/*   o/  tnci-Jt»ct.(d*yi) 

FIG.  26. 


BLADE  SECTIONS,  AND  WORKING  FORMULAE     71 


FIG.  27. 


FIG.  28. 


72 


AIR-SCREWS 


CHAPTEE  V. 

"  LAYING   OUT  "    THE   AIR-SCREW. 

IN  commencing  to  lay  out  the  blade  sections  of  an  air-screw, 
it  is  first  necessary  to  determine  the  chord  angles  of  the  blade 
at  several  radii.  Having  done  this-,  the  plan  and  elevation  of 
the  blade  may  be  drawn  out,  consideration  being  given  to  the 
fact  that  as  far  as  possible  the  two  following  conditions  should 
be  satisfied : — 

(1)  The  centre  of  area  of  the  sections  should   lie  on  the 

blade  axis. 

(2)  The  respective  positions  of  the  centres  of  pressure  of 

the  sections  should  be  so  arranged  about  the  blade 
axis  as  to  eliminate  as  far  as  possible  all  twist  on  the 
blade.  The  loading  on  the  blade  may  be  taken  as 
being  uniplanar. 


FIG.  29. 


These  two  conditions  may  be  occasionally  somewhat 
antagonistic. 

A  symmetrical  plan  form  is  undesirable.  A  good  plan  form 
is  shown  in  Fig.  (29). 


"LAYING  OUT"  THE  AIR-SCKEW  73 

We  can  obtain  the  true  chord   angles  ((/>)  for  each  radius 
f  the  blade  considered  from  the  relation 


(j)v  =  ax  +  tan 
where  (P)  has  the  value  of  —  as  already  denned. 


The  values  of  (ax)  may  vary  along  the  blade,  although 
usually  it  will  be  found  that  the  values  of  the  "  angles  of 
attack  "  are  approximately  constant  and  in  the  neighbourhood 
of  4  degrees. 

If  we  draw,  Fig  (30),  a  vertical  line  to  represent  the  value 
y 
of  —  and  a  horizontal  line  to  represent  (radii  X  2.?r)  in  the  same 

units,  we  may,  by  drawing  in  the  various  hypotenuses,  obtain 
the  inclination  of  the  helix  paths  for  any  element  along  the 
blade.  And  if  these  helix  angles  (A)  be  augmented  by  the 
respective  "  angles  of  attack  "  at  these  points,  we  shall  obtain 
the  true  chord  angles  for  the  various  radii  considered. 

If,  further,  the  widths  of  the  blade  at  these  radii  be  drawn 
in  to  scale  along  the  chord  angle  lines,  we  may  at  once  proceed 
to  lay  out  the  plan  and  elevation  of  the  whole  blade.* 

Sections  near  the  boss  may  be  thickened  up  if  necessary  by 
adding  a  convex  lower  surface.  In  such  sections  the  calculated 
chord  angles  may  have  to  be  departed  from.  This  is  not  of 
great  importance,  although  the  actual  chord  angles  of  such 
sections  should  not  be  less  than  their  respective  helix  angles 
at  these  radii. 

Modern  air-screw  blades  are  built  up  of  several  separate 
laminations  of  wood.  French  walnut  is  usually  chosen  as 
the  most  suitable  material  from  which  to  construct  the 
air-screw. 

The  lamina?  may  be  easily  laid  out  when  the  chord  angles 

*  Strictly  the  blade  widths  and  sections  at  each  radius  when  obtained 
should  lie  on  cylindrical  sections  coaxial  with  the  air-screw,  and  not  on 
plane  sections  at  right  angles  to  a  fixed  arbitrary  line  in  the  blade.  The 
difference,  however,  is  small  at  all  but  the  smallest  radii,  where  it  is  of 
least  importance. 


AIR-SCREWS 


LAYING  OUT"  THE  AIE-SCBEW 


76 


AIE-SCEEWS 


at  the  several  radii  considered  have  been  determined.  The 
method  is  indicated  in  Fig.  (31). 

The  plotting  of  the  contours  along  the  blade  is  obtained 
from  a  consideration  of  the  plan  form  of  blade  and  the  con- 
struction of  the  laminae. '  A  specimen  contour  plotting  is 
shown  in  Fig.  (."32). 

In  the  laying  out  of  the  air-screw  the  various  curves  should 
as  far  as  possible  be  run  into  the  boss  with  smooth  curves,  the 


FIG.  31. 


size  of  the  boss  being  fixed  from  considerations  of  blade  width 
and  type  of  air-screw  mounting  used. 

The  thickening  up  of  blade  sections  by  means  of  a  convex 
under  surface  would  not  appear  to  affect  their  aerodynamic 
properties  to  any  great  extent.  Such  thickening  up  may 
sometimes  be  necessary  from  considerations  of  strength.* 

*  The  employment  of  a  convex  undersurface  appears  to  slightly 
decrease  the  value  of  (cy),  and  hence  necessitates  the  utilisation  of  a 
wider  blade  section.  This  is  sometimes  done  when  a  stronger  section  is 


required.     The 


Lift 
Drag 


ratio  is  only  very  slightly  affected. 


77 


CHAPTEE  VI. 

STRESSES   IN   AIR-SCREAV   BLADES. 

Centrifugal  Stresses. 

WHEN  an  air-screw  is  rotating  about  its  axis  at  the  boss  centre, 
the  various  elements  which  go  to  make  up  each  blade  are  forced 
to  follow  a  circular  path.  The  forces  necessary  to  make  these 
portions  of  the  blade  follow  such  circular  paths  are  directed 
towards  the  point  about  which  the  air-screw  is  rotating  (i.e. 
the  boss  centre),  and  are  of  amount  equal  to 

(weight  of  portion  of  blade  considered)  .  (average  velocity 
of  portion  considered)2  divided  by  (g)  .  (average 
distance  of  portion  from  the  boss  centre). 

Hence  the  reactionary  forces  with  which  the  portions  of  the 
blade  "  pull"  on  any  section  considered  are  of  the  same  amount, 
and  constitute  a  stress  in  the  material  of  which  the  blades  of 
the  air-screw  are  made. 

Let  Fig.  (33)  represent  a  blade  of  an  air-screw,  and  let  AA' 
be  a  section  of  same  at  a  radius  of  (X)  feet  from  the  boss 
centre.  Then  the  centrifugal  stress  011  the  section  AA'  is 
composed  of  the  "  total  pull "  exerted  by  the  portion  of  the 
blade  from  A  A'  to  the  tip,  divided  by  the  area  of  the  section 
at  AA'. 

Let  (?•)  denote  the  overall  length  of  each  blade  in  feet. 
And  consider  the  portion  of  the  blade  from  AA'  to  the  blade 
tip  of  length  (r-X.)  feet. 

Consider  the  centrifugal  pull  on  AA'  due  to  a  small  element 
of  the  blade  cut  off  by  the  two  radii  (#)  and  (x  +  dv) 
respectively.  Then  (X)  remains  constant,  while  (./•)  varies  from 
the  value  (X)  to  the  value  (r). 


78 


AIR-SCREWS 


Then  centrifugal  pull  on  A  A'  due  to  the  element  considered 
may  be  written  , 

/F  _  (weight  of  element)     (velocity  of  element)2 
0  $ 

And  let  (w)  =  weight  in  Ibs.  of  one  cubic  foot  of  the  material 
of  which  the  blades  of  the  air-screw  are  made,  and  (iv)  is  then 
assumed  to  remain  constant  for  all  values  of  (x). 


FIG.  33. 


And  let  (a)  =  area  of  element  considered  in  sq.  ins. 


And  therefore 
in  sq.  ft. 


=  area  °^  sec^on  °f  element  considered 


a.dx 


And  therefore  volume  of  element  =  -Vr  cu.  ft. 
And  therefore  weight  of  element  =  — TJJ-  Ibs. 

And  circumferential  velocity  of  element  is  equal  to 
(2.7r.x.n)  ft./sec. 


STEESSES  IN  AIE-SCEEW  BLADES  79 

So  that  we  may  write 


giving  the  centrifugal  pull  on  AA'  due  to  the  element  con- 
sidered. 

Hence  the  total  pull  on  AA'  is  given  by 

E  =  —  '• — —     a.x.dx  Ibs. 


Whence  the  stress   due  to  the  centrifugal  pull  at  AA'  is 
riven  by 


a.x.dx  Ibs./sq.  in., 


where  (c^)  is  equal   to   the   area   of  the    section   at   A  A'   in 
sq.  ins. 

Now  («)  denotes  the  area  in  sq.  ins.  of  the  section  at 
radius  (#)  from  the  boss  centre,  and  consequently  the  value 
of  (a)  may  vary  with  (x).  It  is,  however,  a  simple  matter  to 

E 

evaluate  graphically  the  expression  for  — ,  if  necessary. 

We  have  then 
Stress  due  to  centrifugal  action  at  any  section  distance  (X) 

F 

from  the  boss  centre  =  —  Ibs./sq.  in., 

where 

P=T  -I' 

=  x 


We    may    at    once    estimate    the    tensile    stress,    due    to 
centrifugal  action,    near    the    boss    of    the    air-screw,   if    we 

O  7 

assume  that 

(1)  The  blade  width  is  uniform,  and 

(2)  The  blade  section  is  constant,  except  near  the  boss. 


80  AIR-SCREWS 

Then  we  have  fit  once  that 

F       7r-.n\w. 


«!  "'    3Q.ff.a-i      alx 
since  («)  is  constant 


And  this  formula  holds  good  for  any  value  of  (X)  providing 
the  initial  assumptions  (1)  and  (2)  are  satisfied. 

Since  near  the  boss  (X)  may  be  taken  equal  to  zero,  the 
formula  becomes 

F       7r2.n2.w.a.r2  ..     , 
-  =  —  ^-r-         -  Ibs./sq.  in., 
«!  7'2.g.al 

and  this  gives  the  value  of  the  tensile'  stress  at  or  near  the  boss 
due  to  the  centrifugal  pull  of  the  whole  blade. 

An  example  will  make  the  application  of  this  result  clearer. 

Let  (n)  =  20  revs./sec.,  (r)  =  4  ft.,  (w)  =  35  Ibs./cu.  ft., 
(a)  =  7  sq.  ins.,  and  («i)  =  16  sq.  ins. 

Then,  taking  the  value  of  (?r2)  as  being  equal  to  10,  we  get 

F       10.400.35.7.16 

«T     ~T2.3216"         ^  lbs./sq.  in. 

Hence  the  amount  of  the  tensile  stress  at  or  near  the  boss, 
due  to  the  centrifugal  pull  of  the  whole  blade,  is  equal  to 
425  Ibs./sq.  in.  This  result  is  quite  in  the  usual  order  of 
practical  work. 

Stresses  due  to  bending. 

The  stresses  in  the  blades  due  to  bending  are  due  to  the 
resultant  air  pressure  exerted  upon  each  element  of  the  blade. 

Consider  any  section  as  in  Fig.  (34). 

And  let  (Yc)  =  the  distance  in  inches  of  the  extreme  ordinate 
of  the  section  from  the  neutral  axis  passing  through  the  centre 
of  area  of  the  section. 

And  let  (Yf)  =  the  distance  in  inches  of  the  chord  line, 
assuming  the  sections  to  be  flat  underneath,  from  the  neutral 
axis. 


STEESSES  IN  AIK-SCEEW  BLADES  81 

Then  the  extreme  values  of  the  tensile  and  compressive 
stresses  occur  at  the  layers  of  the  material  most  remote  from 
the  neutral  axis. 

So  that  we  have 

Maximum  value  of  compression  stress  at  any  section  in 
Ibs./sq.  in.  =  compressive  stress  at  outside  fibres 

=  M.YC 
I 

And  similarly, 

Maximum  value  of  tension  stress  at  same  section  in 
Ibs./sq.  in.  =  tension  stress  at  outside  fibres 


I    ' 


FIG.  34. 

where  (M)  =  bending  moment  at  section  considered,  and 

(I)  =  moment  of  inertia  of  the  area  of  the  section  about 

the  neutral  axis, 
and  (Yc)  and  (Y«)  have  already  been  defined. 

If  the  values  of  — ^-?  and  — '=-*• -  be  then  worked  out  for 

various  values  of  the  radius  (X),  we  can  determine  the  values 

G 


82 


AIR-SCREWS 


of  the  max.  compressive  and  max.  tensile  stresses  due  to  bending 
and  centrifugal  force. 

For  we  have 
Max.  compressive  stress  at  any  section,  due  to  bending  and 

centrifugal  force  =  — ~  -  centrifugal  stress  at  section. 

And  similarly, 
Max.  tensile  stress  at  the  same  section,  due  to  bending  and 


centrifugal  force  = 


M.Y, 


4-  centrifugal  stress  at  section. 


FIG.  35. 


And  if  this  be  done  for  several  sections  of  the  radius  (X} 
we  shall  obtain  the  "  maximum  maximorum "  stress  at  some 
radius,  which  stress  must  not  exceed  the  safe  working  stress  of 
the  material  used  in  the  construction  of  the  air-screw. 

We  have  then  to  determine  the  Bending  Moment  (Mx)  at 
any  distance  (X)  from  the  boss  centre,  that  is,  at  any  section 
considered  along  the  blade. 

Let  Fig.  (35)  represent  a  side  elevation  of  the  air-screw 
blade. 

Consider  a  section  at  A  A'  at  a  distance  (X)  from  the  boss 


STEESSES  IN  AIR-SCREW  BLADES  83 

centre.     Then  the   Bending  Moment  at  AA'  is  equal  to  the 
sum.  of  all  the  B.M.'s  at  the  sections  from  A  A'  to  the  blade  tip. 

Let  (x)  be  any  radius  from  the  boss  centre,  lying  between 
AA'  and  the  blade  tip,  and  let  (dE)  represent  the  resultant  air- 
pressure  at  this  section.  The  value  of  (dE)  has  already  been 
determined  in  terms  of  the  radius  (x). 

Then  B.M.  at  AA'  due  to  the  force  (dE)  =  (dE)  .  (x  -  X), 
and  therefore 
the  total  B.M.  at  AA'  at  a  radius  (X)  from  the  boss  centre 


J»a?  =  r 
dE.  (x  -  X). 
r  =  X 


Now 

(dR)  =  p.tf.b.cy.  (P2  +  4.7T2.a?2).  sec  7.  dx, 
so  that 

^x  -  X)  =  p.n*.  \(x  -  X).  b.Cy.  (P2  +  4.7r2.^2).  sec  7.  dx 

X  Jx  =  X 

=  B.M.  at  AA'  at  a  distance  of  (X)  from  the 

boss  centre. 

We  are  now  able  to  calculate  the  value  of  the  B.M.  on  any 
section  at  any  distance  from  the  boss  centre. 

Since  we  may  without  appreciable  error  take  the  value  of 
(sec  7)  as  equal  to  unity,  the  above  becomes 


;x  =  p.n2. 


x  —  X 


and,  since  (R)  is  usually  measured  in  Ibs.,  (x)  and  (X)  will 
be  measured  in  feet,  when  (p)  =  '00238,  and  therefore  the 
value  of  (Mx)  will  be  obtained  in  Ibs. /ft. 

But,  since  (Yc),  (Yt)  and  (I)  are  usually  measured  in  inches, 
(Mx)  must  also  be  measured  in  inches  in  order  to  give  the 
stresses  in  lbs./sq.  in. 

Hence  we  must  multiply  the  value  of  (Mx),  obtained  from 
the  formula  already  given,  by  (12). 

So  that  we  then  have 
B.M.  at  (X)  in  Ibs./in.  =  (12). (Mx),  which  is  equal  to 


*  f(L- 
Jx  =  X 


G   2 


84  AIK-SCEEWS 

and  we  may  evaluate  this  at  once  if  we  assume  that  the 
conditions  (1)  and  (2)  already  given  are  satisfied,  so  that  (b) 
and  (Cy)  are  constants  over  the  blade. 

To  estimate  the  Bending  Moment  at  or  near  the  boss,  we 
put  (X)  equal  to  zero,  and  the  formula  then  becomes 

B.M.  at  (X)  in  Ibs./in.  =  e>.p.n\l).cy.r\  (P2  -f  2.7r2.r2), 

and  if  we  take  as  an  example  that 

p  =  -00238,  n  =  20,  r  =  4,  b  =  -75,  cy  =  -2,  P  =  5,  ?r2  =  10, 

we  have 

B.M.  in  Ibs./in.  at  or  near  the  boss  is  equal  to 

(6)  (  •  00238)  (400)  (  •  75)  (  -  2)  (16)  (25  +  20x16) 
=  4728  Ibs./in. 

Now  we  have  the  maximum  value  of  Compression  stresses 
at  (X)  ft.  from  the  boss  centre,  due  to  Bending  Moment  and 

centrifugal  force  in  Ibs./sq.  in.  =  12.  —    —  c  —  centrifugal  stress 


=  ~->.?i2.      (*-X).&.<v.(Pi+4.ir»UB«).&! 

ir*.n?.w   (*r       j 
-  -^—          a.x.dx. 
36.0.«i  Jx 


And  similarly  the  maximum  value  of  the  Tension  stresses 
at  (X)  feet  from  the  boss  centre  due  to  Bending  Moment  and 

centrifugal  force  in  Ibs./sq.  in.  =  12.  —  -'  —  -  +  centrifugal  stress 


12.Mx.Yt       £ 

I  «x 


=  12.  Y«  ^2  PL  Xx  5c 

1          ^  ir*n*w    r 

+  T-T — —  •   I  < 

36.tf.ftn  '  Iv 

t/  *          «y/-A 


r 

a.x.  ax. 
x 


And  we  may  ^now  obtain  the  values  of  both  the  maximum 
Tension  and  maximum  Compression  stresses  at  any  distance 
(X)  feet  from  the  boss  centre. 


STKESSES  IN  AIE-SCEEW  BLADES  85 

If  then  the  values  of  these  stresses  be  worked  out  for 
several  values  of  (X),  we  shall  obtain  a  "  maximum  maximorum" 
value  for  the  Tension  and  Compression  stresses  at  some 
particular  value  of  (X). 

The  greatest  of  these  two  values  so  obtained  must  not 
exceed  the  safe  working  stress  of  the  material  of  which  the 
blades  of  air-screws  are  made.  An  approximate  value  for  the 
safe  working  load  of  walnut  is  2000  Ibs./sq.  in. 


86  AIR-SCEEWS 


CHAPTER  VII. 

STATIC   THRUST. 

THE  supposed  "effectiveness"  of  an  air-screw  is  sometimes 
thought  to  depend  upon  the  "  pull "  it  exerts  when  revolving 
on  the  ground,  that  is  when  the  aeroplane  to  which  it  is 
attached  is  being  held  back  prior  to  a  flight. 

Now  although  what  may  be  termed  the  "  Static  "  thrust  of 
an  air-screw  cannot  be  considered  as  necessarily  being  a 
criterion  of  efficiency,  yet  it  is  interesting  to  see  how  far  it  is 
possible  to  predict  quantitative  values  for  this  thrust  when 
considered  in  the  light  of  the  aerofoil  theory. 

It  is  quite  evident  that  the  analogy  still  holds  in  this  case, 
for  we  may  consider  each  section  along  the  blade  as  moving 
with  a  definite  velocity  and  hence  exerting  a  definite  thrust, 
although  the  air-screw  is  acting  like  a  fan,  since  it  has  no 
velocity  in  a  direction  normal  to  the  disc  of  revolution  of  the 
blades.* 

The  helix  angles  (A)  are  thus  equal  to  zero,  and  the 
"  angles  of  attack  "  (a)  must  therefore  be  considered  as  being 
equal  to  the  actual  measured  chord  angles  at  the  various  radii. 

Thus  (P)  =  0,  and  (a)  =  (</>). 

The  general  expression  for  the  thrust  on  an  air-screw  blade 

y 
for  any  value  of  —  is  given  by 

(•r  

T  =  c.p.n*.    f(x).  <j)(x).  (2.TT.X-P.  ^(x)  ).  \XP"-f  4.ira.a;a.  dx. 

Jr0 

*  Owing  to  some  of  the  speeding  up  of  air  occurring  before  the 
air-screw  is  reached,  the  theory  cannot  be  applied  directly  without  very 
appreciable  error.  This  error  is  not  so  great  in  the  case  of  stationary 
air-screws  working  in  an  enclosing  channel. 


STATIC  THKUST  87 

And,  since  in  this  case  (P)  =  0,  this  becomes 

T  =  4.7r2.c.p^2.  P/0).  $(x).  x\  dx, 

Jr0 

And  this  can  be  evaluated  for  any  form  of  air-screw  blade 
considered. 

Let  us  suppose  however  that  the  blade  widths  are  uniform 
from  boss  to  tip,  and  that  the  section  is  also  uniform. 

We  may  regard  (r0)  =  0,  so  that  we  get 


T  =  4.7r2.p.&.?t2.cv.  I  y?.  dx 


and  .-. 


which  gives  an  approximate  value  for  the  "  Static  "  thrust  on 
each  blade. 

The  value  of  (cy)  will  depend  upon  the  form  of  the  blade 
sections  at  the  outside  radii,  and  the  "  average  v  values  of  ($), 
the  chord  angles  of  the  blade  to  the  air-screw's  disc  of 
revolution. 

Approximate  values  for  (cy)  can  be  obtained  from  reference 
to  tests  carried  out  on  sections  similar  in  form  to  those 
employed  at  the  outside  radii  of  the  blade  and  at  angles  of 
the  same  amount  as  the  "  average  "  chord  angles. 

(cy)  often  has  a  value  of  about  (  •  4). 

Ic  is  useful,  however,  to  obtain  this  "  Static  "  thrust  expres- 
sion in  terms  of  the  horse-power  of  the  motor. 

Since 

550.H.5       550.H.7; 
V  P.w, 

in  Ib./ft./sec.  units,  we  obtain 


=     __ 

S.ir.n.d.  tan  7 

The  value  of  (tan  7)  depends  upon  the  values  of  the  outside 


chord  angles. 


88  AIR-SCEEWS 

If  we  take  (tan  y)  =  -,  we  get 
N.T  = 


1 
3' 

700.H 


n.d 

This  formula,  while  being  very  simple,  is  of  course  only  a 
very  approximate  one. 


1000 


8  CO 


GO 


400 


20C 


H.P.  - 


CuYve 


400 


800  /ROC 

Revs,  / 

FIG.  36. 


/600 


8000 


MOO 


If  as  an  example 

H  =  40,  n  =  2Q,d  =  8,  we  get 

XT          700  x  40 

20x8    =  '  a  verv 

In  any  case,  however,  it  would  appear  that  we  can  put  the 
expression  for  the  "  Static  "  thrust  into  the  form 


N.T  = 


X.H 

n.d' 


where  (X)  has  a  value  in  the  neighbourhood  of  (1000). 


STATIC  THEUST 


Suppose 

H  =  40,  n  =  8,  d  =  8,  \  =  1000 ;  then 

XTT        1000x40 
N'T=       8x8 

We  notice  that  the  "  Static "  thrust  increases  inversely  as 
the  rate  of  revolution  of  the  air-screw,  so  that  we  should  expect 
a  slow-running,  i.e.  geared-down,  air-screw  to  have  a  higher 


26OC 

Sfahc  Te»f(V"c). 

ROOD 

<n 
-Q 
^ 
•^  /500 

i/l 

3 

>. 
-C 

K 

TArusf  -  5/>rcd  Cur^e 
Projbe//er, 

/ 

/ 

/ 

500 

/ 

/ 

_^ 

/ 

o 

400                 80O               '200               /600               t.  000                2*00 

??evs.  per  Min. 

FIG.  37. 

"  Static "  thrust  than  one  which  was  coupled  direct  to  the 
motor,  for  the  same  horse-power. 

The  "  Static "  thrust  of  an  air-screw  on  an  aeroplane  is 
usually  measured  by  attaching  a  spring  balance  to  the  rear 
portion  of  the  machine  and  attaching  a  rope  from  the  spring 
balance  to  some  fixed  support. 

We  see  also  from  the  general  formula  for  the  static  thrust 
of  an  air-screw  that  the  static  thrust  varies  as  the  square  of  the 


90  AIR-SCBEWS 

rotational  speed,  and  that  the  necessary  B.H.P.  required  to  turn 
the  air-screw  varies  as  the  cube  of  the  rotational  speed. 

This  is  borne  out  approximately  by  experimental  tests  on 
air-screws,  as  Figs.  (36),  (37)  will  show.  These  Figures  are 
taken  from  the  results  of  tests  carried  out  at  the  National 
Physical  Laboratory  by  Mr.  F.  H.  Brain  well  and  Mr.  A.  Fage.* 

We  notice  from  these  experimental  curves  that  for  a  static 
thrust  of  approximately  1050  Ibs.  on  an  8  ft.  diameter  air- 
screw having  a  rotational  speed  of  1600  revs,  per  min.,  the 
B.H.P.  necessary  is  approximately  270.  Hence  if  we  apply 
the  formula  deduced  for  the  static  thrust  we  may  determine 
an  approximate  value  for  the  constant  (X). 

We  have 

"X  TT 
Total  static  thrust  of  air-screw  =  X.T  = 

n.d 

whence 

X.270.60 
1600.8  ' 

and  therefore 

(X)  =  830. 

This  value  is  rather  smaller  than  the  value  proposed,  from 
reference  to  actual  static  tests  carried  out  on  aeroplane  air- 
screws, for  this  constant,  namely  1000. 

*  See  "  The  Aeroplane,"  By  A.  Fage,  A.E.C.Sc.  (Charles  Griffin  &  Co.). 


91 


CHAPTER   VIII. 

EFFICIENCY   OF   AN   AIR-SCREW   AT   DIFFERENT    SPEEDS    OF 
TRANSLATION. 

PRESENT  day  aircraft,  whether  aeroplanes  or  dirigibles,  have  as 
a  rule  two  distinct  limiting  speeds  of  flight.  In  the  dirigible 
the  minimum  speed  will  of  course  be  zero,  but  in  the  aeroplane 
this  is  not  so,  and  the  minimum  possible  speed  of  flight  for 
any  given  type  may  be  calculated  approximately  when  the 
characteristics  of  the  machine  are  known. 

Now  between  these  two  limiting  speeds,  that  is  between  the 
minimum  and  maximum  climbing  speeds,  there  will  be  an 
infinite  range  of  speeds  at  which  the  aeroplane  may  fly.  That 
is  to  say  that,  if  an  aeroplane  has  a  maximum  velocity  of  say 
100  feet  per  second  and  a  minimum  velocity  of  say  50  feet 
per  second,  then  between  these  two  outside  speeds  the  aeroplane 
has  an  infinite  'number  of  different  velocities  at  which  it 
may  fly. 

Now  it  will  usually  be  found  that  in  the  case  of  an  aeroplane 
the  speed  at  which  it  is  able  to  climb  fastest,  that  is  the  speed 
at  which  the  reserve  thrust  horse-power  of  the  motor  is  greatest, 
will  lie  between  its  maximum  and  minimum  speeds  and  will 
usually  be  nearer  the  latter. 

If  we  examine  the  thrust  horse-power  curve  of  an  aeroplane 
it  will  usually  be  found  that  the  ordinates  on  the  curve  have 
at  some  value  of  the  abscissae  (in  this  case  velocity)  a  minimum 
value,  and  the  velocity  corresponding  to  this  minimum  point 
will  be  the  velocity  at  which  the  thrust  horse-power  required 
for  horizontal  flight  is  a  minimum,  and  hence  the  reserve  thrust 
horse-power  is  a  maximum,  that  is  the  velocity  of  ascent  will 
be  greatest  at  this  value  of  the  velocity. 


92  AIE-SCEEWS 

Now  this  is  only  true  when  the  efficiency  of  the  air-screw 
is  supposed  to  remain  constant  throughout  the  interval  between 
the  maximum  and  minimum  speeds  of  flight.  In  practice, 
however,  this  is  never  so,  since  the  air-screw  is  usually  designed 
for  the  maximum  flight  velocity,  and  hence  for  any  other  value 
of  the  velocity  the  efficiency  of  the  air-screw  falls  off. 

Hence  the  curve  of  available  thrust  horse-power  is  now  no 
longer  a  straight  line,  and  therefore  the  maximum  difference 
between  the  ordinates  of  this  curve  and  the  curve  of  thrust 
horse-power  required  for  horizontal  flight  will  no  longer  neces- 
sarily occur  at  the  velocity  corresponding  to  the  minimum 
value  of  the  thrust  horse-power  required  for  horizontal  flight, 
but  will  usually  be  found  to  occur  at  a  value  of  the  flight 
velocity  slightly  greater  than  this  value. 

Now  in  order  to  be  able  to  determine  this  point  on  the 
curve  and  hence  the  maximum  rate  of  climb  possible  and  also 
the  speed  of  flight  corresponding  to  this  maximum  climb,  it  is 
necessary  to  know  how  the  efficiency  of  the  air-screw  varies 
for  different  values  of  (V),  the  velocity  of  flight.  Experimental 
results  of  tests  on  air-screws  have  been  obtained  for  various 
types  and  curves  of  efficiency  plotted  against  values  of  (V). 

It  is  interesting,  however,  to  attempt  to  predict  the  amount 
of  this  variation  in  efficiency  from  the  results  already  obtained 
on  the  assumption  of  the  aerofoil  analogy. 

We  have  already  obtained  an  expression  for  the  efficiency 

y 
of  any  type  of  air-screw  at  any  value  of  the  effective  pitch  — , 

tv 

that  is  at  any  value  of  the  velocity  of  advance  (V). 

Hence  we  may,  providing  we  possess  the  necessary  informa- 
tion with  regard  to  the  values  of  the  lift  and  j —  of  the  sections 

drag 

at  various  angles  of  incidence,  determine  the  value  of  (r;),  the 
efficiency  of  the  air-screw,  under  varying  sets  of  conditions  and 
for  various  values  of  the  translational  velocity. 

Now  it  has  already  oeen  shown  that  the  results  obtained 
from  the  "  Rational^'  blade  form  in  many  cases  approximate 
very  closely  to  standard  types  of  air-screw  blades,  and  we  shall 
therefore  use  this  form  of  blade  in  the  quantitative  determina- 


EFFICIENCY  AT  DIFFEKENT  SPEEDS  93 

tion  of  the  values  of  (7;)  for  various  values  of  the  velocity  of 
advance  (V). 

This  modification  is  introduced  for  convenience  and  simplicity 
of  working  out  the  numerical  examples,  which  become  very 
tedious  when  treated  from  the  most  general  expression  for 
efficiency. 

We  have  then  the  efficiency  of  an  air-screw  of  this  type  is 
given  by 


P.    (  X.  ftx).  (2.7T.X  -  P.  ^(X)  ).  dx 
Jr0 

J,r 
•x*.  $0).  (P  +  2.TT.X.  ^r(x)  ).  dx 
>•„ 


for  any  value  of  (P)  =  —  ,  and  we  proceed  to  discuss  the  varia- 

tion in  the  value  of  (77)  for  variations  in  the  value  of  (V). 

Now  it  is  not  proposed  to  attempt  to  evaluate  algebraically 

y 
the  expression  for  (77)  for  any  value  of  —  ,  since  to  do  so  would 

71 

necessitate  the  determination  of  at  least  approximate  equations 

y 
for   the   two   functions   <f>(x)   and   ty(x)  for   each  value   of  - 

considered. 

We  shall  therefore  employ  a  graphical  method  in  this 
investigation. 

It  will  be  necessary  to  determine  the  value  of  (77),  the 
efficiency  of  the  air-screw,  for  several  values  of  the  advance  per 

revolution  —  .  and  when   these  values   have  been  obtained   a 
n 

smooth  curve  drawn  through  the  points  will  give  at  any  rate 
a  near  approximation  to  the  value  of  the  air-screw's  efficiency 
at  any  value  of  the  velocity  (V)  considered. 
We  have  then  to  plot  the  two  graphs  of 


and 


Y 

for   each   value   of  —   taken,  then   determine   the   two   areas 

n 


94  AIK-SCKEWS 

enclosed  by  each  respectively,  the  extreme  ordinates  at  radii  of 
(?'0)  and  (r),  and  the  (x)  axis,  divide  the  area  thus  enclosed  by 
the  first  curve  by  the  area  enclosed  by  the  second  curve,  and 

V 

multiply  the  result   by    ^ ,  where   (V)  has  the  particular 

value  of  the  velocity  of  advance  chosen.     The  value  of  (P)  in 

y 

each  case  will  then  of  course  be  — ,  (V)  having  the  particular 

rfi 

value  already  defined. 

Since  we  know  the  form  of  the  blade  sections  at  various 
radii  we  may,  by  reference  to  tests  carried  out  in  a  wind- 
channel  on  sections  of  similar  form,  determine  the  probable 
values  of  (cy)  and  (tan  7),  that  is  of  </>(x)  and  ^(#),  of  the 
sections  at  these  radii  when  we  know  the  respective  values  of 
the  angles  of  attack  of  the  sections  considered. 

We  may  determine  the  values  of  these  angles  of  attack  for 

y 

each  value  of  (V),  and  hence  of  — ,  chosen  as  follows. 

iii 

We  must  first  determine  the  values  of  the  chord  angles  (</>) 
at  the  radii  considered,  and  we  can  do  this  either  by  direct 
measurement  of  these  angles  on  the  air-screw,  or,  if  we  know 

Y 

for   what   value   of  -  -  the   air-screw  is    designed,  it   is  only 
id 

necessary  to  determine  the  values  of  (</>)  for  any  radius  chosen 
from  the  relation 


tan 


-' 


where  (P)  has  the  value  of  the  designed  effective  pitch,  that  is 
the  value  of  the  velocity  of  the  aircraft  (usually  the  maximum 
velocity)  for  which  the  air-screw  is  designed,  divided  by  the 
value  of  (n)  at  this  velocity. 

(ax)  is  the  value  of  the  "  angle  of  attack  "  of  a  section  at 
radius  (x),  for  the  velocity  of  the  aircraft  for  which  the  air- 
screw is  designed. 

(ax)  may  be  termed  the  "initial  angle  of  attack"  of  any 
section  at  radius  (x). 

Having  then  found  the  value  of  (fa),  the  chord  angle  for 


EFFICIENCY  AT  DIFFERENT  SPEEDS 


95 


any  radius  (x),  we  may  proceed  to  determine  the  new  value  of 
(ax)  for  the  various  values  of  (V)  considered. 

Since  the  effective  pitch  has  now  a  different  value  from  its 
"initial"  designed  value,  we  may  denote  it  by  (P'),  and  the 
new  "  angles  of  attack "  at  each  radius  (x)  by  (axr).  We  then 
have  at  once  that 

«;  =  <^- tan-' (£-, 

and  we  are  now  in  a  position  to  make  the  following  table  for 
each  value  of  *(V)  taken. 


Section  No. 

7 

6 

5 

4 

Eadius  (x)  feet  . 

1  foot 

2  feet 

3  feet 

4  feet 

Chord    angle    at) 
radius  (x)  .      ./ 

42°  30' 

25  3  42' 

18°  51' 

15°  15' 

Angle   of  attack) 
at  radius  (x)     / 

10° 

8° 

7o 

6°  13' 

Abs.  lift  coeff.  at) 
radius  (x)  .      J 

•338 

•525 

•475 

•425 

1 

1 

1 

1 

radius  (x)  . 

^8 

3-4 

10-5 

12 

The  above  table  has  been  taken  as  an  example.  It  is  for  an 
air-screw  designed  for  a  velocity  of  advance  (V)  of  100  feet 
per  sec.,  and  a  speed  of  revolution  (n)  of  20  revs,  per  second. 
The  value  of  the  effective  pitch  is  thus  5  feet,  and  the  table 
given  is  for  the  case  considered  when  the  velocity  of  advance 
is  80  feet  per  second,  so  that  the  new  value  of  the  effective 
pitch  is  4  feet.  The  sections  are  those  given  in  the  "  Technical 
Eeport  of  the  Advisory  Committee  for  Aeronautics,  1911-12," 
page  76,  and  already  referred  to  in  a  previous  chapter. 

The  values  of  <f>(x)  and  -^(x)  are  taken  from  the  curves 
plotted  as  a  result  of  tests  in  a  wind-channel  on  these  sections 
at  the  National  Physical  Laboratory. 

Similar  tables  to  the  one  given  may  now  be  obtained  for 
y 

several  other  values  of  —  from  zero  up    to  and  beyond  the 

n 


96  AIK-SCBEWS 

designed  velocity  of  flight.     From  each  of  these  tables  we  may 
plot  two  curves,  the  curves  respectively  of 


and 

&.#X).  (P  +  2.TTJCW)  ), 

and  the  two  areas  enclosed  by  these  two  curves  may  then  be 

found  in  each  case.     We  may  thus  calculate  the  efficiency  of 

y 

the  air-screw  for  each  value  of  -  chosen,    and   the   values   so 

n 

obtained  can  then  be  plotted  against  their  respective  values  of 
(V),  giving  a  curve  of  efficiency  for  any  value  of  the  velocity  of 
advance  of  the  air-screw  considered. 

It  will  be  found  that  when  (V)  =  0,  then  (?;)  =  0,  and  that 
when  (V)  has  values  appreciably  greater  than  the  designed 
value  of  (V)  the  value  of  (rj)  rapidly  falls  off. 

An   example   will   make    the   application   of  this  method 
clearer. 
Let 

(V)  the  designed  or  initial  value  of  the  flight  velocity 
of  the  aircraft  =  100  ft.  per  second. 

(n)  the  speed  of  revolution  of  the  air-screw  at  this 
value  of  (V)  =  20  revs,  per  second. 

Then       (P)  the  initial  value  of  the  effective  pitch  =  5  feet. 
(d)  the  diameter  of  the  air-screw  =  8  feet. 

(ax)  the  initial  angles  of  attack  of  the  sections  along 
the  blade  =  4°  =  constant  over  blade. 

Let  the  sections  along  the  blade  be  those  already  referred  to 
and  let  them  be  spaced  as  follows  :  — 

At  a  radius  of  1  foot,  section  no.  7. 
„      2  feet,        „        „    6. 

O  K 

»          »       o     ,,  ,,        ,,     o. 

»         „      4     „  „        „    4. 

We  may  then  form  the  following  table  :  — 


EFFICIENCY  AT  DIFFERENT  SPEEDS  97 

y 

-  =  5  feet. 


Section  No. 

7 

6 

5 

4 

^ 

42°  30' 

25°  42' 

18°  51' 

15°  15' 

ax 

4° 

4° 

4° 

4° 

(j>(x) 

•416 

•384 

•374 

•344 

1 

1 

1 

1 

Y(x) 

10-2 

11-4 

11-8 

12-8 

X 

Ifoot 

2  feet 

3  feet 

4  feet 

From  this  table  plot  the  two  graphs  of 


and 


taking  values  of  <£(#)  and  ^r(x)  for  each  value  of  (x)  taken  from 
the  above  table.     (P)  has  of  course  the  value  of  (5). 

These  two  curves  are  shown  plotted  in  Fig.  38. 

The  efficiency   of  the  air-screw   under  these    conditions  is 

seen  to  be  75%. 

Y 
Now  take  successive  values  of  —  ,  of  say  1  ft.,  2  ft.,  3  ft.,  4  ft., 

fv 

6  ft.,  and  7  ft.,  and  in  each  case  form  the  corresponding  table. 
These  tables  are  given  below  in  order. 


-  =  1  foot. 
n 

Section  No. 

7 

6 

5 

4 

0* 

42°  30' 

25°  42' 

18°  51' 

15°  15' 

ax 

33°  28' 

21°  9' 

15°  49' 

13° 

#(») 

•44 

•41 

•53 

*(*) 

•• 

1 
2-2 

1 

3-1 

1 

7 

X 

Ifoot 

2  feet 

3  feet 

4  feet 

II 


98 


AIR-SCREWS 


G9-  - 

{A-reo     rnclo»ed    by    TopCurve     = 
w  M  Lower     .       = 


13  38 
20  63 


...    -n    =        _  .  -    =    .75 

1  2.TT         SO    63 


35 


15 


2' 

R  a  d  i  u  s(x)/«« 
FIG.  38. 


EFFICIENCY  AT  DIFFERENT  SPEEDS 


99 


Y 

-  =  2  feet. 

n 

Section  No. 

7 

6 

5 

4 

*. 

42°  30' 

25°  42' 

18°  51' 

15°  15' 

ax 

25° 

16°  40' 

12°  48' 

10°  42' 

*(*) 

•48 

•39 

•425 

•575 

1 

1 

1 

1 

YW 

2 

2  -.5 

4-3 

10 

03 

1  foot 

2  feet 

3  feet 

4  feet 

Y 

-  =  3  feet. 

Section  No. 

7 

6 

5 

4 

0* 

42°  30' 

25°  42' 

18°  51' 

15°  15' 

o» 

17° 

12°  18' 

9°  49' 

8°  27' 

*(«) 

•38 

•38 

•56 

•5 

1 

1 

1 

1 

y(aO 

2-3 

3'7 

9 

11 

* 

1  foot 

2  feet 

3  feet 

4  feet 

Y 

-  =  6  feet. 

n 

Section  No. 

7 

6 

5 

4 

0, 

42°  30' 

25°  42' 

18°  51' 

15°  15' 

ax 

-  1°  11' 

o°ir 

1°13' 

1°48' 

*(«) 

•25 

•26 

•275 

•25 

1 

1 

1 

1 

VW 

7'5 

9 

10-5 

11 

a? 

Ifoot 

2  feet 

3  feet 

4  feet 

H    2 


100 


AIR-SCKEWS 


y 

=  7  feet. 
n 


Section  No. 

7 

6 

5 

4 

0* 

42°  30' 

25°  42' 

18°  51' 

15°  15' 

ax 

-  5°  30' 

-  3°  20' 

-  1°  27' 

-  0°  21' 

MX) 

•06 

•12 

•17 

•15 

, 

1 

1 

1 

1 

Y(x) 

1-5 

b-5 

6 

7 

X 

Ifoot 

2  feet 

3  feet 

4  feet 

From  these  tables  plot  the  curves  given  in  Figs.  39,  40,  41, 
42,  43,  and  44. 

The  respective  efficiencies  are  seen  to  be  as  follows. 

The  value  of  (n)  has  been  supposed'  to  remain  constant 
throughout.  This  assumption  does  not  affect  in  any  way  the 
generality  of  the  method. 

y 

At  a  speed  of  translation  of  20  ft./sec.,  -  =  1  ft.,  (rj)  =  15-5% 

11 

„  40  „  „  =2,,  „  =  32-6% 

„  60  „  „  =3,,  „  =  54-5% 

„  80  „  „  =  4  „  „  =  67% 

„  120  „  „  =6,,  „  =  76-7% 

»  140  „  „  =  7,,  „  =62-7% 

A  curve  of  efficiency  against  values  of  (V)  may  now  be 
plotted,  Fig.  (45). 

Y 

It  will  be  noticed  that  when  —  =  6  ft.,  i.e.  at  a  velocity  of 

translation  of  120  ft.  per  second,  the  efficiency  of  the  air-screw 
is  higher  than  the  efficiency  at  the  speed  for  which  it  is 
designed.  This  is  due  to  the  fact  that  the  value  of  (Z), 

i.e.  — ,  is  greater  than  the  initial  value,  and,  although  the  -^~ 
a  drag 

of  the  individual  sections  is  less  than  in  the  initial  case,  yet 


EFFICIENCY  AT  DIFFERENT  SPEED'S          101 


osed      >^    To\a    Curve        s.       2.S-82   »q 

.    Low«v    .        =     E3-5       - 


FIG.  39. 


102 


AiE-SCEEWS 


FIG.  40. 


EFFICIENCY  AT  DIFFEKENT  SPEEDS          103 


50 


45 


10 


GO- 


JAv. 


AYCQ    enclosed     bj  Top  Curve.    =     86 '14 

.    Lower    .         =    £8   66 


a' 


FIG.  41. 


104 


AIK-SCBEWS 


I  Ar«o   enc/ostfd      by    Top  Curve     -   E4-34  s<j,  mv 
.     Lowev     %        -    £3  /4 

-       .       _         _        V  Area  enclosed    by  Tof>  Curve 

p  TT  |Q         /\v€a  e nclose^^^Tl^^v^M^^^I we 

g>  £4-34 

TT  S3     I  4 


67 


"Radius  (x) 

FIG.  42. 


EFFICIENCY  AT  DIFFERENT  SPEEDS          105 


50 


35 


fl»*o    enclosed     byTo».Curv«        »        O55      S^   .oS.l 
.    Lower     .          »      f6-84     ,.      .     ) 


&  .-.    T 


"  si?  • 


-767 


R  Qd/ 


FIG.  43. 


106 


AIR-SCEEWS 


{b-rca     enclosed    by  Top  Curve        =      7. 
.     Lower    .          =•    13 


.54 
4 


g.TT 


Rod  i  u  <,  (x)  /« 
FIG.  44. 


EFFICIENCY  AT  DIFFEEENT  SPEEDS          107 


OR)'"'- 

FIG.  45. 


108  AIK-SCEEWS 

the  higher  value  of  (Z)  makes  a  preponderance  in  the  efficiency 

y 

of  the   whole   blade.     At  a  value  of  -  -  =7  ft.,  however,  we 

n 

notice  that  the  efficiency  is  very  much  smaller  in  spite  of  the 
still  higher  value  of  (Z). 

Of  course  if  the  air-screw  was  designed  initially  for  a  value 

V 

of     -  =  6  ft.,  the  efficiency  would   be  greater  than  that  now 

y 

given  at  this  value  of  — ,  providing  of  course  that  the  spacing 

/TL 

and  forms  of  the  sections,  etc.,  remained  the  same  as  in  the 
initial  case  here  considered. 

This  curve  of  efficiency  against  values  of  translational 
velocity  (Y)  appears  to  conform  to  curves  obtained  by 
experiment  on  similar  types  of  air-screws. 


109 


CHAPTEE  IX. 

DIRECT    LIFTING    SYSTEMS. 

WHEN"  a  force  (P)  moves  its  point  of  application  through  a 
distance  (s),  in  the  direction  of  the  force,  it  is  said  to  do  (P.,s) 
units  of  work. 

If  a  weight  of  (W)  Ibs.  be  lifted  through  a  distance  of 
(x)  feet  the  work  done  against  gravity  is  (W.x)  ibot-lbs.,  sup- 
posing the  distance  (x)  to  be  negligible  in  comparison  with  the 
earth's  radius. 

Now  suppose  that  a  body  of  weight  (W)  Ibs.  is  being  lifted 
against  gravity  at  a  uniform  speed  of  (Y)  feet/sec.,  then  the 
work  done  per  second  is  equal  to  (W.V)  ibot-lbs.,  and  the 
horse-power  necessary  to  keep  the  body  moving  at  this  velocity 


is  equal  to  -550  . 

And  suppose  the  B.H.P.  of  the  motors  installed  in  the 
machine  to  be  denoted  by  (H),  and  the  efficiency  of  the  lifting 
screws  to  be  (rj),  then  the  Effective  Thrust  Horse-Power  of  the 
motors  is  equal  to  (H.??),  and  this  is  the  horse-power  available 
for  keeping  the  machine  in  motion. 

So  that  we  have 

11-77  =     550 ' 
whence 

V  =  — ,i?'—  feet/sec., 
W 

or 

W.V 

:    550^' 


110  AIK-SCKEWS 

Now  suppose  that  (V)  =  0,  so  that  the  machine  remains 
stationary  in  the  air,  then 

H  ------ 

=  550'  rj   =  :  V 

and  if  (77)  is   not   zero,   the    B.H.P.   required   to   sustain   the 
weight  (W)  vanishes. 

But  from  the  general  expression  for  the  efficiency  of  any 
type  of  air-screw,  we  notice  that  when  the  effective  pitch, 
that  is  the  advance  per  revolution  of  the  air-screw,  is  zero 
the  efficiency  of  the  air-screw  is  also  zero,  so  that  (H) 
is  not  necessarily  a  vanishing  quantity,  for  we  obtain  the 
expression 

W_   Q        ^ 

~  550*0  "     0' 

which  is  indeterminate,  and  may  be  zero  or  a  finite  quantity. 

We  can  however  obtain  the  value  of  (H)  for  the  case 
(V)  =  0  from  a  consideration  of  the  more  general  case  when 
(V)  is  not  zero. 

And  if,  having  obtained  the  value  of  (H)  for  finite  values 
of  (V),  we  consider  (V)  and  therefore  the  effective  pitch  to 
become  very  small  and  ultimately  vanish,  we  shall  obtain  the 
required  value  of  (H)  for  the  case  when  the  machine  remains 
stationary  in  the  air. 

Suppose  then  that  a  body  is  moving  vertically  upwards 
with  a  uniform  velocity  of  (V)  feet/sec.,  under  the  action  of 
screws  which  revolve  at  (n)  re  vs. /sec.  Then  the  effective  pitch 

y 
of  such  screws  will  be  -•  feet,  and  if  we  know  the  diameter  (d) 

71* 

of  each  screw  we  can  find  an  expression  for  the  efficiency  of 
the  screws  in  terms  of  the  quantities  already  known. 

We  will  suppose  for  the  sake  of  simplicity  that  we  are 
using  a  shape  of  air-screw  blade  similar  to  that  already  defined 
as  the  "  Eational "  blade.  Then,  if  we  further  suppose  that 
(cy)  and  (tan  7)  are  constants  over  the  blade,  and  that  (tan  7) 

has  the  value  of  =-^,  we  can  at  once  write  down  an  expression 


DIRECT  LIFTING  SYSTEMS  111 

for  the  efficiency  of  the  whole  air-screw  under  the  conditions 
specified. 

The  above  assumptions  are  made  for  the  sake  of  simplicity 
and  convenience,  and  do  not  affect  the  general  results  to  any 
very  great  extent.  The  "  Rational "  form  of  blade  outline  is 
such  a  simple  one  to  deal  with  analytically  that  the  great 
simplification  in  the  work  obtained  by  using  this  form  of  blade 
is  a  justification  for  not  treating  the  problem  of  direct  lift  in 
its  more  general  form,  when  the  blade  outline  is  considered  as 
some  arbitrary  undetermined  function  of  the  radius  (x). 

The  expression  for  the  efficiency  is  therefore  given  by 

2.Z.(8.7r-Z) 
•^.(TT  +  IG.Z)' 

P        V 

and,  since  (Z)  =  -7  =  — «  we  obtain 

Cv  ??/ .  Ct> 


And  this  expression  is  the  value  of  the  efficiency  of  the 
lifting  screws  under  the  conditions  specified. 

We  are  now  in  a  position  to  estimate  the  necessary  B.H.P. 
of  the  motors  for  the  velocity  (V)  upwards.  We  have  already 
shown  that  the  required  B.H.P.  is  given  by 

II  -     W     V 

"  550*  7,' 

and 

2. V.  (S.TT.n.d  -V) 


71    = 

so  that 


1100'        (S.Tr.n.a-  \) 

• 

W  TT  nd 
And  now,  if  (V)  =  0,  then  (H)  =  —^^-t 


which  gives  the  B.H.P.  necessary  for  "  hovering." 


112  AIR-SCEEWS 

This  was  the  case  which  before  we  were  unable  to  evaluate 
owing  to  the  indeterminateness  of  the  expression-.,. 
Thus  as  an  example  suppose  that 

W  =  1100  Ibs., 
n  =  20  revs. /sec., 
d  =  8  feet, 

then  the  B.H.P.  necessary  to  keep  the  machine  hovering  in  the 
air  is  given  by  the  value  of  (H),  which  in  the  above  example  is 
found  to  be  equal  to  (63)  B.H.P.  approx. 

It  can  be  seen  from  the  formula  that  when  (V)  is  not  zero, 
the  B.H.P.  required  is  greater  than  what  it  was  for  the  case 
when  (V)  =  (0),  and  as  (V)  increases  so  does  (H)  increase. 

The  value  of  (H)  is  strictly  the  value  of  the  B.H.P.  of  the 
motors  multiplied  by  the  efficiency  of  the  transmission  to  the 
lifting  screws. 

It  will  be  noticed  that  when 

(V)  =  8.7r.n.d 

the  value  of  (H)  becomes  infinite,  which  forms  therefore  the 
limiting  value  of  (V).  This  limiting  value  of  (V)  is  caused 
by  the  fact  that  when  (V)  has  the  value  given  above,  (Z)  is 
equal  to  (8.77-.),  and  this  corresponds  to  a  zero  value  of  the 
efficiency  of  the  air-screws. 

Now  in  order  that  the  screws  may  be  capable  of  lifting 
the  whole  machine,  it  is  necessary  that  their  combined 
effective  thrusts  shall  be  greater  than  the  total  weight  of  the 
machine  (W). 

This  is  equivalent  to  saying  that  the  air-screws  must  be 
capable  of  sustaining  a  greater  weight  than  (W)  when  (V)  =  0, 
for  if  it  be  supposed  that  the  machine  has  an  extra  weight  (w) 
attached  to  it,  and  that  the  screws  are  just  supporting  the 
combined  weights  of  (W)  and  (w)  in  the  air,  then  it  is  obvious 
that  if  the  weight  (w)  be  detached  from  the  machine  it  will 
fall,  and  the  machine  will  then  be  subjected  to  an  accelerating 
force  greater  than  its  weight  (W),  and  hence  that  it  wTill 
commence  to  rise. 


DIEECT  LIFTING  SYSTEMS  113 

In  order  therefore  that  our  helicopter  may  be  capable  of 
rising  off  the  ground,  the  effective  horse-power  of  the  motors 
must  satisfy  the  relation 

TJ-       W.ir.n.d 
*>     880CT- 

The  quantitative  determination  of  the  acceleration  produced 
by  the  reserve  horse-power  is  somewhat  difficult  to  arrive  at. 

There  are  however  other  considerations  bearing  upon  the 
subject  of  vertical  ascent  in  the  air. 

We  have  already  shown  that 


H          W 

~  1100 


y  =  Tr.n.d.(8800.TI-W.7r.n.d) 


whence 


which  gives  the  velocity  of  ascent  through  the  air. 
We  notice  at  once  that  when 

W.Tr.n.d 
8800 

the  value  of  (Y)  is  zero,  and  the  machine  remains  stationary. 
This  is  the  condition  already  established  for  "hovering"  flight. 
•    Iii  order,  therefore,  that  (V)  may  be  positive,  it  is  necessary 
that 

W.ir.n.d 
8800    ' 

and  this  condition  brings  us  to  a  consideration  of  the  requisite 
values  of  (n)  and  (d),  which  values  so  far  have  been  assumed 
to  be  anything  whatever. 

It  is  of  course  obvious  that  in  order  to  obtain  a  good 
efficiency  for  the  lifting  screws,  the  value  of  (Z)  should  approxi- 
mate to  the  value  at  which  the  efficiency  of  the  screws  is  a 
maximum.  But  since  it  will  be  found  that  the  velocity  of 
ascent,  and  hence  the  effective  pitch,  may  be  small  compared 


114  AIPt-SCEEWS 

to  the  diameter  of  the  air-screw,  unless  this  diameter  be  itself 
small,  it  will  be  more  economical  to  use  possibly  a  large 
number  of  separate  air-screws,  in  which  the  ratio  of  effective 
pitch  to  diameter  is  such  as  to  entail  at  least  a  moderately 
good  efficiency.  This  may  of  course  make  the  diameter  of 
each  individual  air-screw  quite  small.  The  reason  for  using  a 
number  of  separate  air-screws  is  that  the  necessary  width  of 
blade  would  be  inordinately  large  in  the  case  of  one  or  two 
lifting  screws  only,  and  hence  the  value  of  (c)  would  be  such 
as  to  entail  an  enormous  amount  of  interfering  action  between 
the  blades  of  each  air-screw. 

Suppose  then  that  there  are  (Q)  separate  air-screws,  each 

W 

exerting  a  thrust  of  -~-  Ibs.,  then  the  horse-power  available  for 

turning  each  screw  is  ^-. 

And  let  each  air-screw  (all  of  which  are  assumed  to  be 
identical)  have  (N)  blades. 
Then  we  have 

Total  thrust  exerted  by  each  air-screw  =  (N.T)  Ibs., 

and 

Total  weight  required  to  be  supported  by  each  air-screw 

=    Q  lbS" 

whence 

W 
(N.T)  =--  Q, 

Now  we  have  for  simplicity  assumed  that  the  blade  shape 
for  each  air-screw  is  that  defined  as  the  "Eational"  blade 
shape.  And  we  have  further  assumed  that  the  section  of  the 
blade  is  uniform  throughout  the  entire  blade,  except  perhaps 
near  the  boss,  which  does  not  affect  the  argument  to  any 
appreciable  extent.- 

We  can  then  at  once  apply  the  formula  already  obtained 
for  the  thrust  of  an  air-screw  having  this  shape. 


DIEECT  LIFTING  SYSTEMS  115 

We  have 

c.^^p.cy.l\d'2.(2.7r.d-3.P^nry) 

12 
whence 

/  V  \ 

~rT        c.n.TT.p.V.d^.Cu.  I  2.7T.d  —  3.  —  - .  tan  7  ) 
W  V  n          '/ 

Q   :  ~l2~  ' 

and  /. 

12.W 


Q  = 


c.p.7T.V.d2.cy.  (2.7r.d.n  —  3.V.  tan  7)' 


In  this  formula  for  the  determination  of  (Q)  all  the  factors 
are  known  with  the  exception  of  (V),  the  velocity  of  ascent. 
We  have,  however,  already  obtained  a  formula  for  (V),  viz. 

ir.n.d.(88QQ.'E-W.Tr.n.d) 

~  ' 


so  that  we  can  at  once  determine  the  least  necessary  value  of 
(Q)  for  any  chosen  value  of  (c). 
Suppose  that 

W  =  1000  Ibs., 

n  =  20  revs.  /sec., 

d  =  10  feet, 
H  =  100  effective  horse  power, 

and  let 

1 
C  =  3' 

Cy      =        "4, 

tan  7  =  j^, 

then,  since  we  are  using  lb./ft./sec-  units,  (p)  will  have  the 
value  of  (-00238),  and  hence  we  get  the  value  of  (Q)  as 
given  by 

12x1000 

y  =  -i  --  ~r~  1?: 

i  X  •  00238  XTTX  15  x  102  x  '4^2.^.10.20  -  ~ 

I  2 


116  AIR-SCREWS 

/ 

We  thus  obtain   the  least  value  of  (Q)  required  in  order 
that  the  initial  conditions  may  be  satisfied. 

We  notice  that  if  (c)  be  less  than  -^  the  necessary  value 

o 

of  (Q)  will  be  greater  than  (6). 

If  we  solve  formally  to  obtain  (Q)  in  terms  of  (H),  we  get 

*    = 


and  it  can  be  seen  from  this  that  as  (d)  decreases  in  value  (Q) 
increases.  It  is  interesting  to  consider  for  what  value  of  (d) 
(V)  has  a  maximum  value,  and  hence  to  obtain  the  necessary 
A'alue  of  (Q)  for  this. 

To  find  the  value  of  (d)  for  which  (V)  is  a  maximum,  we  put 

Id   =  °' 

and  this  gives 

_  (712-5).H       227.H 

w.v.^r      WM 

as  the  value  giving  (V)  a  maximum  value. 

To  obtain  the  value  of  the  maximum  value  of  (V),  we 
substitute  the  value  of  (d)  obtained  from  the  above  in  the 
general  formula  for  (Y),  and  get 

462.H  , 
vmax  =       w    feet/sec., 

and  this  gives  the  maximum  velocity  upwards  under  the  best 
possible  conditions. 

The  value  of  the  number  (462)  depends  upon  the  value  of 
(tan  7)  taken. 

We   also   notice   that   the   efficiency  of  the  lifting  screws 

under  these  conditions  is  equal  to  ^-^,  that  is  approximately 


(84)  per  cent. 

Thus  a  machine,  weighing  complete  with  engines,  screws 


DIRECT  LIFTING  SYSTEMS  117 

fuel,  etc.,  (1000)  Ibs.,  and  having  motors  capable  of  developing 
an  effective  horse-power  of  (100),  would  have  an  upward 
velocity  under  the  best  conditions  as  already  given  of  (46  •  2) 
feet  per  second,  or  roughly  (2800)  feet  per  minute.  This  rate 
of  climbing  is  about  double  that  of  the  fastest  aeroplane  scouts 
at  present  in  existence. 

We  can  now  obtain  the  necessary  value  of  (Q)  for  a 
maximum  value  of  (V). 

We  get 


QV    *JiJj.j\*-v        )-   »»    ./i./fc 
7     ~r    T-f*  J 

and  since 

227  H 

d  =     ^r'  :    for  a  maximum  value  of  (Y), 
W.n 

where  (n)  is  arbitrary,  we  can  find  the  necessary  value  of  (Q) 
for  any  arbitrary  chosen  value  of  (d). 
This  then  gives 

o  =  (_'i99)ao-4).w_3 

whence  the  larger  the  value  of  (d)  the  smaller  the  value  of  (Q). 
Applying  this  result  to  the  previous  example,  where 

W  =  1000  Ibs., 
_  1 

<'<!/    =     *3> 

H  -  100, 
d  =  10  feet, 


we  get 


Q  =  26-2. 


So  that  we  shall  require  at  least  (27)  separate  helices  in 
order  that  (c)  may  have  a  value  not  greater  than  -,. 

Thus  the  blades  would  have  to  be  between  two  and  three 


118  AIR-SCREWS 

feet  wide  and  the  air-screw  would  have  to  rotate  at  a  speed 
given  by  the  relation 

227.H 
W.d' 
so  that 

(n)  =  2*  27,  which  is  approximately  (136)  revs./rnin. 

It  is  a  simple  matter  to  determine  the  necessary  value  of 
the  blade  widths  at  any  radius  (x)  along  the  blade.     We  have 

2.C.P.^.. 


giving  the  value  of  (b)  for  any  radius  (x). 

At  the  tip  of  the  blade  (x)  =  (r),  and  the  value  of  (b)  is  then 
given  by 

C.l\7T.d 

br  =  - 


Applying   this   last   result   to   the    example   given    above 


1        46-2 

-        X 


we  get 


So  that  if  N  =  2,  that  is,  if  each  air-screw  has  (2)  blades 
we  get 

br  =  (2  '65)  feet  (approx.), 

and  this  is  the  necessary  value  of  the  blade  width  at  the  tip 
under  the  conditions  specified. 


As  stated  in  the  Preface  the  results  given  by  the  theory  in 
the  case  of  Direct  Lifting  Systems  and  for  Static  Thrust  must 
not  be  accepted  without  due  caution. 

Owing  to  the  fact  that  in  both  these  cases  the  translational 
velocity  of  the  machine  is  usually  zero  or  very  small,  and 
hence  the  ratio  of  the  slip-stream  velocity  to  this  velocity 
rather  high,  the  angles  of  attack  of  the  various  blade  elements 


DIEECT  LIFTING  SYSTEMS  119 

are  not  the  same  as  given  by  the  theory,  and  consequently  the 
results  obtained  from  it  by  calculation  may  be  quite  fallacious. 

It  would  appear,  however,  that,  providing  the  value  of  (Z) 
is  sufficiently  large  and  in  the  neighbourhood  of  that  obtaining 
in  standard  types  of  air-screws  as  used  upon  aeroplanes  (that  is 
having  a  value  of  say  between  *4  and  unity),  the  results  as 
given  by  the  theory  in  Chap.  IX.  should  be  found  to  be 
sufficiently  true,  at  any  rate  as  a  basis  for  practical  design  and 
further  investigation. 

This  would  probably  necessitate  a  very  high  rate  of  ascent, 
or  at  any  rate  the  utilization  of  the  value  of  (Q)  giving  a 
maximum  value  to  (V),  in  order  to  obtain  the  requisitely  large 
value  for  (Z). 

The  calculated  value  of  the  B.H.P.  required  for  "  hovering," 
given  on  page  112,  is  for  these  reasons  probably  much  too  small. 

Mr.  F.  W.  La.nchester,  in  a  recent  paper,*  gives  a  formula 
for  the  determination  of  the  least  requisite  B.H.P.  for  a 
stationary  helicopter,  his  expression  being 


where  (A)  is  the  area  of  the  propeller  disc,  (W)  is  the  weight 
sustained,  and  (p)  the  density  of  the  fluid,  in  the  case  of  air 

approx.  —  ,  or  0  •  078. 

If  we  apply  this  to  the  example  quoted,  we  obtain  (135)  as 
a  minimum  value  for  (H).  This  is  probably  a  much  more 
accurate  figure. 

*  "The  Screw  Propeller,"  by  F.  W.  Lanchester,  M.Inst.C.E.,  read 
before  the  Institution  of  Automobile  Engineers  in  April  1915. 


121 


APPENDIX   I 

NOTE   ON   THE   INFLUENCE   OF   "ASPECT   EATIO." 

IN  the  Introduction  a  brief  reference  was  made  to  the  ratio  of 
the  span  to  the  chord  of  a  wing,  commonly  known  as  the 
"  Aspect  Batio."  It  is  fairly  certain  that  the  characteristics  of 
an  aerofoil  vary  with  variation  of  aspect  ratio,  although  any 
exact  quantitative  determination  of  the  alteration  of  lift  and 

,  corresponding  to  a  given  change  in  the  value  of  the 
drag' 

aspect  ratio  of  a  wing,  would  appear  to  be  impossible  at 
present. 

At  the  same  time  it  would  appear  that  an  increase  in  the 

,       lift     , 
value  of  the  -^ — -  always  accompanies  an  increase  in   aspect 

ratio. 

The  graph  given  in  Fig.  46  is  plotted  from  the  result  of 
tests  on  a  wing  carried  out  at  the  National  Physical  Labora- 
tory,* and  serves  as  an  indication  of  the  kind  of  change  to  be 
expected.  The  difficulty  of  correctly  anticipating  the  amount 
of  this  change  in  any  case  appears  to  be  largely  due  to  the 
form  of  the  wing  tips  employed. 

This  question  of  aspect  ratio  should  be  taken  into  account 
as  far  as  possible  when  designing  an  air-screw,  as  experimental 
tests  have  shown  that  high  efficiencies  may  usually  be  expected 
from  screws  having  blades  of  a  high  aspect  ratio. 

In  fact  it  is  apparent  that,  without  attempting  to  formulate 
any  exact  connection  between  blade  efficiency  and  aspect  ratio, 
a  high  speed  air-screw  might  conceivably  have  a  better 
efficiency  owing  to  the  necessary  comparative  narrowness  of 

*  "  Technical  Eeport   of  the  Advisory  Committee   for  Aeronautics, 
1911-12." 


122 


AIR-SCREWS 


the  blades  than  one  designed  for  the  same  conditions  but  to 
rotate  at  a  slower  and  otherwise  more  economical  speed. 

It  is  obvious,  therefore,  that  this  question  cannot  be 
altogether  neglected  either  in  aeroplane  or  air-screw  design, 
and,  in  view  of  the  comparatively  meagre  information  at 
present  available  on  the  subject,  further  experimental  research 
in  this  direction  would  appear  to  be  required. 


/  2  3  4 


&  10 


FIG.  46. 


123 


APPENDIX    II 

NOTE  ON  THE  EFFECT  OF  THE  INDRAUGHT  IN  FRONT 
OF  AN  AIR-SCREW. 

IT  was  stated  in  the  Preface  that  the  theory  here  outlined  was 
not  in  any  sense  to  be  regarded  as  bearing  any  more  than  a 
fairly  close  relationship  to  the  actual  conditions  surrounding 
the  working  of  a  screw  in  air.  It  would  indeed  be  well  nigh  an 
impossibility  to  formulate  a  theory  which  would  adequately 
deal  with  all  the  various  complex  factors  entering  into  the 
problem  of  screw  propulsion  in  fluids,  and  the  most  that 
scientific  analysis  can  do  is  to  build  up  some  kind  of  a  working 
hypothesis  which  may  reasonably  be  expected  to  give  results 
sufficiently  true  for  the  purpose  of  practical  design. 

In  fact,  in  any  investigation  of  this  kind,  certain  factors 
bearing  upon  the  problem  may  have  to  be  ignored  owing  to  the 
difficulty  of  accurately  representing  their  effects  without  the 
too  continuous  employment  of  experimental  data  in  the  form 
of  checks  upon  the  theory. 

One  such  factor,  of  which  no  quantitative  notice  has  been 
taken  in  the  previous  work,  is  the  indraught  of  air  in  front 
of  an  airscrew,  the  effect  of  which  is  to  modify  the  conditions 
under  which  any  element  of  blade  has  been  assumed  to  be 
working.  The  main  modifications  introduced  by  such  an 
indraught  appear  to  be  a  decrease  in  the  "  angle  of  attack  "  and 
an  increase  in  the  relative  air  velocity  of  each  element  along 
the  blade,  thus  altering  to  some  extent  the  values  of  the  air 
reactions  upon  the  same,  although  it  is  evident  that  these  two 
effects  tend  to  partially  neutralise  each  other.  Quantitatively, 
however,  it  appears  to  be  very  difficult  to  readjust  the  foregoing 
theory  so  as  to  include  this  effect  of  indraught,  without  very 
much  greater  experimental  evidence  than  is  at  the  present 
time  available.  Some  experiments  have  been  carried  out  by 
G.  Eiffel  *  to  determine  the  magnitude  of  this  incoming  air, 
and  he  shews  that  the  ratio  of  the  indraught  velocity  to  the 

velocity  of  translation  is  a  function  of  the  =¥•        -  ratio  only. 

diameter 

*  "  Nouvelles  Recherches  sur  la  Resistance  de  1'Air  et  1'Aviation,"  1914. 


124  AIB-SCBEWS 

If  we  assume  that  the  analogy  with  aerofoils  still  holds  for 
every  element  of  the  blade,  it  is  a  simple  matter  to  shew  that 
this  ratio  cannot  exceed  a  certain  limit  directly  depending 
upon  the  value  of  the  "  angle  of  no  lift  "  for  the  blade  section 
considered.  The  value  of  this  ratio  for  air-screws  of  the  type 

as  at  present  employed,  and  having   —5         _  ratios  of  from 

-   diameter 

(  •  5)  to  (  *  7),  appears  to  lie  in  the  neighbourhood  of  (  •  6)  to  (  •  7) 
for  the  effective  portion  of  the  blade. 

Further,  if  we  assume  that  the  correction  factor  already 
given  for  the  calculated  torque  of  a  blade  to  be  capable  of 
being  applied  to  each  element  along  the  blade  between  the 
limits  of  integration,  we  obtain  a  value  of  about  (  •  4)  for  this 
ratio  as  sufficient  to  account  for  the  difference  between  the 
values  of  the  torque  as  calculated  and  found  by  experiment. 


The  value  of  the  correction  factor  taken  is  (-^ 

\7o  J 

For  a  value  of  (  •  4)  for  this  ratio,  the  "  angles  of  attack  "  of 
the  elements  near  the  blade  tip  are  found  to  be  between  (3°) 
and  (4°)  less  than  the  original  values  assigned  to  them  when 
no  account  was  taken  of  the  indraught.  This  would  make  the 
real  values  of  these  angles  of  attack  nearly  zero  for  blades 
designed  on  the  aerofoil  theory. 

Eiffel's  experiments  do  not,  however,  appear  to  substantiate 
this  view,  the  values  of  this  ratio  for  various  speeds  of  trans- 
lation appearing  not  to  exceed  (-1).  In  any  case,  without 
further  experimental  evidence  it  is  quite  useless  to  attempt  to 
fix  any  definite  value  for  this  ratio. 

This  question  of  the  ratio  of  the  indraught  velocity  to  the 
translational  velocity  is  one  which  becomes  of  fundamental 
importance  in  the  case  of  screws  having  very  small  or  zero 
values  of  the  effective  pitch  to  diameter  ratios,  and  were  it 
possible  to  obtain  an  accurate  determination  of  the  amount  of 
this  ratio  in  such  cases,  the  problem  of  the  helicopter  already 
discussed  would  be  much  easier  of  a  representative  solution 
than  has  so  far  been  found  to  be  the  case,  owing  to  the 
limitations  of  the  present  theory. 


125 


INDEX 


"  Aerial  Flight,"  3 
Aerodynamical     efficiency, 

48 

Aerofoil,  1 

Aeronautical  Society,  40 
"  Aeroplane,"  90 
Air-flow,  1 
Air  pressure,  16 
"  Air-screws,"  Paper  on,  40 
"Angles  of  attack,"  13,  56 
Angles  of  incidence,  13 
"  Aspect  Ratio,"  3,  121 
"  Average  "  reaction,  11,21 

Bending  moment,  81,  82 
Bent  plane,  1 
B.H.P.,  18,  52,  59,  119 
Biplane,  analogy  of,  30 
Blade  elements,  13 
Blade  sections,  14,  47 
Blade  shape,  25 
Blade  tip  angle,  10 
Blade  width  constant,  59 


Bolas,  H.,  65 

Bramwell,  F.  H.,  4,  12,  90 

Bryan,  G.  H.,  1 

(c),  value    of,    for    normal 

incidence,  3 

Calculus  of  Variations,  55 
Camber,  48 
Centre  of  area  of  sections, 

72 

Centre  of  pressure  of  sec- 
tions, 72 

Centrifugal  forces,  5,  77 
Centrifugal  pull,  77 
Chord  angles,  10,  13 
4 '  Constant  Pitch,"  10 
"Constructional       Limit" 

type  of  blade,  35 
Contours,  plotting  of,  76 
Convex  under-surface,  76 
Correction  factor  on  blade 

width,  63 
Cylinder,  7 


126 


AIR-SCKEWS 


Diameter  of  air-screw,  51 


Fluid       motion,       discon- 
tinuous, 1 

Formula  for  air-pressure,  1 
Formulae,  working.  47 


Direct  lifting  systems,  109, 

118 

Dirigible,  91 
Discontinuous    blade    out-  |  Fundamental  hypothesis,  3 

line  curve,  44 
Discontinuous     fluid    mo-     "Gap,"  31 

tion,  1  |  "Gap/chord"  ratio,  31 

Drag,  2  j  Gliding  angle,  3,  14 

Drag  coefficient,  absolute,  2  i  Graphical  methods,  iv,  60, 
Drzewiecki,  iii,  3,  4 


"Effective  Pitch,"  9,  10 
Efficiency  curve,  27 


Helicoidal  path,  6 
I  Helicopter,  113,  119 


"  Efficiency    curve  "    blade  j  Helix,  7,  8 

shape,  29,  35  j  Helix,  angle  of,  9 

Efficiency,  maximum  point  I  Helix  angles,  13 

of,  27 


Efficiency  of  an  air-screw, 

19,  91 
Efficiency  of  an  element,  26 


Helix,  length  of,  8 
Hovering  flight,  113,  119 

Ideal    thrust   grading  dia- 


Eiffel,  G.,  123  gram,  43 

Elliptical  shaped  ends,  55       Indraught  in  front  of   an 

Empirical  multiplying  fac- 

tor,  63 
Equations,  17 


air-screw,  123 
Infinitesimals,  23 
Integration,  15 
Experimental  Mean  Pitch,  !  Interference,  30 

12,  19,21 
Experimental  research,  51    |  J°nes>  &»  * 


Fage,  A.,  4,  90 
Flat  plate,  1 


Laminae,  34,  73 
Lanchester,  F.  W.,  3,  4, 119 


INDEX 


127 


Laying  out  blade,  72 
Lift  coefficient,  absolute,  2 
Lift/drag  ratio,  2 
Lift  of  an  aerofoil,  2 
"  Load  Grading  Curve,"  23 
Low,  A.  R.,  34,  38,  40 

Marine  propellers,  4 

Marine  work,  iii 

Mathematical  theory,  1 

Mathematics,  iv 

"  Maximum  maximorum," 
82,  85 

Maximum  ordinate,  posi- 
tion of,  27,  30 

Moment  of  inertia,  8 1 

National  Physical  Labora- 
tory, 4,  90 

Negative  pressure,  2 

Neutral  axis,  80 

"Normale"  blade  shape, 
34,  35 

Outside  fibres,  81 

Pitch  of  an  air-screw,  6 
Pitch   of  zero    "  average " 

Reaction,  11 
Pitch  of  zero  Thrust,  1 1 
Pitch  of  zero  Torque,  1 1 
Pitch  ratio,  40 


Pressure,  negative,  2 
Pressure,  positive,  2 

"  Rational "  blade  efficiency 

formula,  40 
"  Rational "    blade    shape, 

34,  35 
Resultant  air  pressure,   1, 

14,  21,  22 

Resultant  Thrust,  1 1 
Resultant  Torque,  11 
Royal  Society,  Proceedings 

of,  1 

Screws,  marine,  4 
Slip-stream,  43 
Slip-stream  velocity,  119 
Speeding  up  of  air,  86 
Static  Thrust,  86,  118 
Stresses,  47 
Stresses,  bending,  80 
Stresses,  centrifugal,  77 
Symmetrical  plan  form,  72 

"  Technical  Report  of  the 
Advisory  Committee  for 
Aeronautics,  1911-12," 
49,  65 

"  Technical  Report  of  the 
Advisory  Committee  for 
Aeronautics,  1912-13," 
4,  69 


128  AIK-SCREWS 

"  The  Screw  Propeller,"  119  i  Velocity,  minimum,  91 
Thickness/chord  ratio,  47,  ;  Velocity  of  blade  element, 

56  16 

Thrust  of  a  blade,  18,  19      i  Vertical  ascent,  113 
Thrust  of  an  element,  1 6 

Thrust,  Static,  86  Walnut,  safe  working  load 

Torque  of  a  blade,  18  of,  85 

Torque  of  an  element,  17        "Wash"    of     top     wing, 
Translational    velocity,    9,         30 

118  Whitehead,  A.  N.,  iv 

I  Wind-tunnel,  19 

Velocity,  climbing,  91  Working  formulae,  47 

Velocity,  maximum,  91         ;  Wright  Aeroplane,  65 


LONDON  :    PRINTED  BY  WILLIAM   CLOWES   AND  SONS,   LIMITED, 
DUKE  STREET,   STAMFORD  STREET,   S.E.,   AND  GREAT  WINDMILL  STREET,   W. 


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